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FUNCTIONS

PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Tue, 25 Jan 2011 11:35:56 UTC

ContentsArticlesAdditive function Algebraic function Analytic function Antiholomorphic function Arithmetic function Bijection Binary function Bochner measurable function Bounded function Cauchy-continuous function Closed convex function Coarse function Completely multiplicative function Concave function Constant function Continuous function Convex function Differentiable function Doubly periodic function Elementary function Elliptic function Empty function Entire function Even and odd functions Flat function Function of a real variable Function composition Functional (mathematics) Harmonic function Hermitian function Holomorphic function Homogeneous function Identity function Implicit and explicit functions 1 3 6 9 10 20 23 24 25 26 27 27 28 29 31 32 41 45 47 48 49 51 52 54 57 58 58 61 63 68 69 72 75 76

Indicator function Injective function Invex function List of types of functions Locally integrable function Measurable function Meromorphic function Monotonic function Multiplicative function Multivalued function Negligible function Nowhere continuous function Periodic function Piecewise linear function Pluriharmonic function Plurisubharmonic function Polyconvex function Positive-definite function Proper convex function Pseudoconvex function Quasi-analytic function Quasiconvex function Quasiperiodic function Radially unbounded function Rational function Real-valued function Ring of symmetric functions Simple function Single-valued function Singular function Smooth function Subharmonic function Sublinear function Surjective function Symmetrically continuous function Quasisymmetric function Transcendental function Unary function

79 82 85 86 87 89 91 93 97 99 102 103 104 107 109 110 112 112 114 115 117 118 121 122 122 126 126 132 133 133 134 140 143 144 147 147 150 152

Univalent function Vector-valued function Weakly harmonic function Weakly measurable function

152 153 158 158

ReferencesArticle Sources and Contributors Image Sources, Licenses and Contributors 159 163

Article LicensesLicense 164

Additive function

1

Additive functionIn mathematics the term additive function has two different definitions, depending on the specific field of application. In algebra an additive function (or additive map) is a function that preserves the addition operation: f(x + y) = f(x) + f(y) for any two elements x and y in the domain. For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. In number theory, an additive function is an arithmetic function f(n) of the positive integer n such that whenever a and b are coprime, the function of the product is the sum of the functions: f(ab) = f(a) + f(b). The remainder of this article discusses number theoretic additive functions, using the second definition. For a specific case of the first definition see additive polynomial. Note also that any homomorphism f between Abelian groups is "additive" by the first definition.

Completely additiveAn additive function f(n) is said to be completely additive if f(ab) = f(a) + f(b) holds for all positive integers a and b, even when they are not co-prime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f(1) = 0. Every completely additive function is additive, but not vice versa.

ExamplesExample of arithmetic functions which are completely additive are: The restriction of the logarithmic function to N. The multiplicity of a prime factor p in n, that is the largest exponent m for which pm divides n. a0(n) - the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n (sequence A001414 [1] in OEIS). For example: a0(4) = 2 + 2 = 4 a0(20) = a0(22 5) = 2 + 2+ 5 = 9 a0(27) = 3 + 3 + 3 = 9 a0(144) = a0(24 32) = a0(24) + a0(32) = 8 + 6 = 14 a0(2,000) = a0(24 53) = a0(24) + a0(53) = 8 + 15 = 23 a0(2,003) = 2003 a0(54,032,858,972,279) = 1240658 a0(54,032,858,972,302) = 1780417 a0(20,802,650,704,327,415) = 1240681 The function (n), defined as the total number of prime factors of n, counting multiple factors multiple times, sometimes called the "Big Omega function" (sequence A001222 [2] in OEIS). For example; (1) = 0, since 1 has no prime factors (20) = (225) = 3 (4) = 2

Additive function (27) = 3 (144) = (24 32) = (24) + (32) = 4 + 2 = 6 (2,000) = (24 53) = (24) + (53) = 4 + 3 = 7 (2,001) = 3 (2,002) = 4 (2,003) = 1 (54,032,858,972,279) = 3 (54,032,858,972,302) = 6 (20,802,650,704,327,415) = 7 Example of arithmetic functions which are additive but not completely additive are: (n), defined as the total number of different prime factors of n (sequence A001221 [3] in OEIS). For example: (4) = 1 (20) = (225) = 2 (27) = 1 (144) = (24 32) = (24) + (32) = 1 + 1 = 2 (2,000) = (24 53) = (24) + (53) = 1 + 1 = 2 (2,001) = 3 (2,002) = 4 (2,003) = 1 (54,032,858,972,279) = 3 (54,032,858,972,302) = 5 (20,802,650,704,327,415) = 5 a1(n) - the sum of the distinct primes dividing n, sometimes called sopf(n) (sequence A008472 [4] in OEIS). For example: a1(1) = 0 a1(4) = 2 a1(20) = 2 + 5 = 7 a1(27) = 3 a1(144) = a1(24 32) = a1(24) + a1(32) = 2 + 3 = 5 a1(2,000) = a1(24 53) = a1(24) + a1(53) = 2 + 5 = 7 a1(2,001) = 55 a1(2,002) = 33 a1(2,003) = 2003 a1(54,032,858,972,279) = 1238665 a1(54,032,858,972,302) = 1780410 a1(20,802,650,704,327,415) = 1238677

2

Additive function

3

Multiplicative functionsFrom any additive function f(n) it is easy to create a related multiplicative function g(n) i.e. with the property that whenever a and b are coprime we have: g(ab) = g(a) g(b). One such example is g(n) = 2f(n).

See also Sigma additivity

Further reading Janko Brai, Kolobar aritmetinih funkcij (Ring of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp. 97108) (MSC (2000) 11A25)

References[1] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa001414 [2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa001222 [3] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa001221 [4] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa008472

Algebraic functionIn mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials. For example, an algebraic function in one variable x is a solution y for an equation

where the coefficients ai(x) are polynomial functions of x. A function which is not algebraic is called a transcendental function. In more precise terms, an algebraic function may not be a function at all, at least not in the conventional sense. Consider for example the equation of a circle:

This determines y, except only up to an overall sign:

However, both branches are thought of as belonging to the "function" determined by the polynomial equation. Thus an algebraic function is most naturally considered as a multiple valued function. An algebraic function in n variables is similarly defined as a function y which solves a polynomial equation in n+1 variables:

It is normally assumed that p should be an irreducible polynomial. The existence of an algebraic function is then guaranteed by the implicit function theorem. Formally, an algebraic function in n variables over the field K is an element of the algebraic closure of the field of rational functions K(x1,...,xn). In order to understand algebraic functions as functions, it becomes necessary to introduce ideas relating to Riemann surfaces or more generally algebraic varieties, and sheaf theory.

Algebraic function

4

Algebraic functions in one variableIntroduction and overviewThe informal definition of an algebraic function provides a number of clues about the properties of algebraic functions. To gain an intuitive understanding, it may be helpful to regard algebraic functions as functions which can be formed by the usual algebraic operations: addition, multiplication, division, and taking an nth root. Of course, this is something of an oversimplification; because of casus irreducibilis (and more generally the fundamental theorem of Galois theory), algebraic functions need not be expressible by radicals. First, note that any polynomial is an algebraic function, since polynomials are simply the solutions for y of the equation

More generally, any rational function is algebraic, being the solution of

Moreover, the nth root of any polynomial is an algebraic function, solving the equation

Surprisingly, the inverse function of an algebraic function is an algebraic function. For supposing that y is a solution of

for each value of x, then x is also a solution of this equation for each value of y. Indeed, interchanging the roles of x and y and gathering terms,

Writing x as a function of y gives the inverse function, also an algebraic function. However, not every function has an inverse. For example, y = x2 fails the horizontal line test: it fails to be one-to-one. The inverse is the algebraic "function" . In this sense, algebraic functions are often not true functions at all, but instead are multiple valued functions. Another way to understand this, which will become important later in the article, is that an algebraic function is the graph of an algebraic curve.

The role of complex numbersFrom an algebraic perspective, complex numbers enter quite naturally into the study of algebraic functions. First of all, by the fundamental theorem of algebra, the complex numbers are an algebraically closed field. Hence any polynomial relation p(y, x) = 0 is guaranteed to have at least one solution (and in general a number of solutions not exceeding the degree of p in x) for y at each point x, provided we allow y to assume complex as well as real values. Thus, problems to do with the domain of an algebraic function can safely be minimized.

Algebraic function

5

Furthermore, even if one is ultimately interested in real algebraic functions, there may be no adequate means to express the function in a simple manner without resorting to complex numbers (see casus irreducibilis). For example, consider the algebraic function determined by the equation

Using the cubic formula, one solution is (the red curve in the accompanying image)

A graph of three branches of the algebraic function y, where y3xy+1=0, over the domain 3/22/3 < x < 50.

There is no way to express this function in terms of real numbers only, even though the resulting function is real-valued on the domain of the graph shown. On a more significant theoretical level, using complex numbers allow one to use the powerful techniques of complex analysis to discuss algebraic functions. In particular, the argument principle can be used to show that any algebraic function is in fact an analytic function, at least in the multiple-valued sense. Formally, let p(x,y) be a complex polynomial in the complex variables x and y. Suppose that x0C is such that the polynomial p(x0,y) of y has n distinct zeros. We shall show that the algebraic function is analytic in a neighborhood of x0. Choose a system of n non-overlapping discs i containing each of these zeros. Then by the argument principle

By continuity, this also holds for all x in a neighborhood of x0. In particular, p(x,y) has only one root in i, given by the residue theorem:

which is an analytic function.

MonodromyNote that the foregoing proof of analyticity derived an expression for a system of n different function elements fi(x), provided that x is not a critical point of p(x,y). A critical point is a point where the number of distinct zeros is smaller than the degree of p, and this occurs only where the highest degree term of p vanishes, and where the discriminant vanishes. Hence there are only finitely many such points c1, ..., cm. A close analysis of the properties of the function elements fi near the critical points can be used to show that the monodromy cover is ramified over the critical points (and possibly the point at infinity). Thus the entire function associated to the fi has at worst algebraic poles and ordinary algebraic branchings over the critical points. Note that, away from the critical points, we have

since the fi are by definition the distinct zeros of p. The monodromy group acts by permuting the factors, and thus forms the monodromy representation of the Galois group of p. (The monodromy action on the universal covering space is related but different notion in the theory of Riemann surfaces.)

Algebraic function

6

HistoryThe ideas surrounding algebraic functions go back at least as far as Ren Descartes. The first discussion of algebraic functions appears to have been in Edward Waring's 1794 An Essay on the Principles of Human Knowledge in which he writes: let a quantity denoting the ordinate, be an algebraic function of the abscissa x, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of x, and then find the integral of each of the resulting terms.

References Ahlfors, Lars (1979). Complex Analysis. McGraw Hill. van der Waerden, B.L. (1931). Modern Algebra, Volume II. Springer.

Analytic functionThis article is about both real and complex analytic functions. The article holomorphic function is solely about analytic functions in complex analysis. An analytic signal is a signal with no negative-frequency components. In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if it is equal to its Taylor series in some neighborhood of every point.

DefinitionsFormally, a function is real analytic on an open set D in the real line if for any x0 in D one can write

in which the coefficients a0, a1, ... are real numbers and the series is convergent to (x) for x in a neighborhood of x0. Alternatively, an analytic function is an infinitely differentiable function such that the Taylor series at any point x0 in its domain

converges to (x) for x in a neighborhood of x0. The set of all real analytic functions on a given set D is often denoted by C(D). A function defined on some subset of the real line is said to be real analytic at a point x if there is a neighborhood D of x on which is real analytic. The definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane."

Analytic function

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ExamplesMost special functions are analytic (at least in some range of the complex plane). Typical examples of analytic functions are: Any polynomial (real or complex) is an analytic function. This is because if a polynomial has degree n, any terms of degree larger than n in its Taylor series expansion will vanish, and so this series will be trivially convergent. Furthermore, every polynomial is its own Maclaurin series. The exponential function is analytic. Any Taylor series for this function converges not only for x close enough to x0 (as in the definition) but for all values of x (real or complex). The trigonometric functions, logarithm, and the power functions are analytic on any open set of their domain. Typical examples of functions that are not analytic are: The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulas in different regions) are typically not analytic where the pieces meet. The complex conjugate function is not complex analytic, although its restriction to the real line is the identity function and therefore real analytic.

Alternate characterizationsIf is an infinitely differentiable function defined on an open set equivalent. 1) is real analytic. 2) There is a complex analytic extension of to an open set 3) For every compact set there exists a constant non-negative integer k the following estimate holds: which contains . and every , then the following conditions are

such that for every

The real analyticity of a function at a given point x can be characterized using the FBI transform. Complex analytic functions are exactly equivalent to holomorphic functions, and are thus much more easily characterized.

Properties of analytic functions The sums, products, and compositions of analytic functions are analytic. The reciprocal of an analytic function that is nowhere zero is analytic, as is the inverse of an invertible analytic function whose derivative is nowhere zero. (See also the Lagrange inversion theorem.) Any analytic function is smooth, that is, infinitely differentiable. The converse is not true; in fact, in a certain sense, the analytic functions are sparse compared to all infinitely differentiable functions. For any open set C, the set A() of all analytic functions u:C is a Frchet space with respect to the uniform convergence on compact sets. The fact that uniform limits on compact sets of analytic functions are analytic is an easy consequence of Morera's theorem. The set of all bounded analytic functions with the supremum norm is a Banach space. A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function has an accumulation point inside its domain, then is zero everywhere on the connected component containing the accumulation point. In other words, if (rn) is a sequence of distinct numbers

Analytic function such that (rn)=0 for all n and this sequence converges to a point r in the domain of D, then is identically zero on the connected component of D containing r. Also, if all the derivatives of an analytic function at a point are zero, the function is constant on the corresponding connected component. These statements imply that while analytic functions do have more degrees of freedom than polynomials, they are still quite rigid.

8

Analyticity and differentiabilityAs noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or C). (Note that this differentiability is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions which are not analytic: see non-analytic smooth function. In fact there are many such functions, and the space of real analytic functions is a proper subspace of the space of smooth functions. The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that any complex function differentiable (in the complex sense) in an open set is analytic. Consequently, in complex analysis, the term analytic function is synonymous with holomorphic function.

Real versus complex analytic functionsReal and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts.[1] According to Liouville's theorem, any bounded complex analytic function defined on the whole complex plane is constant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by

Also, if a complex analytic function is defined in an open ball around a point x0, its power series expansion at x0 is convergent in the whole ball. This statement for real analytic functions (with open ball meaning an open interval of the real line rather than an open disk of the complex plane) is not true in general; the function of the example above gives an example for x0=0 and a ball of radius exceeding1, since the power series 1 x2 + x4 x6... diverges for |x|>1. Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of the complex plane. However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function (x) defined in the paragraph above is a counterexample, as it is not defined for x=i.

Analytic function

9

Analytic functions of several variablesOne can define analytic functions in several variables by means of power series in those variables (see power series). Analytic functions of several variables have some of the same properties as analytic functions of one variable. However, especially for complex analytic functions, new and interesting phenomena show up when working in 2 or more dimensions. For instance, zero sets of complex analytic functions in more than one variable are never discrete.

Notes[1] Krantz, Steven; Harold R., Parks (2002), A Primer of Real Analytic Functions (Second ed.), Birkhuser, ISBN 0817642641, ISBN0-8176-4264-1, 3-7643-4264-1

References Conway, John B. (1978). Functions of One Complex Variable I (Graduate Texts in Mathematics 11). Springer-Verlag. ISBN0-387-90328-3. Krantz, Steven; Harold R., Parks (2002), A Primer of Real Analytic Functions (Second ed.), Birkhuser, ISBN 0817642641, ISBN0-8176-4264-1, 3-7643-4264-1

External links Weisstein, Eric W., " Analytic Function (http://mathworld.wolfram.com/AnalyticFunction.html)" from MathWorld. Analytic Functions Module by John H. Mathews (http://math.fullerton.edu/mathews/c2003/ AnalyticFunctionMod.html)

Antiholomorphic functionIn mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of the complex variable z defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to z* exists in the neighbourhood of each and every point in that set, where z* is the complex conjugate. One can show that if f(z) is a holomorphic function on an open set D, then f(z*) is an antiholomorphic function on D*, where D* is the reflection against the x-axis of D, or in other words, D* is the set of complex conjugates of elements of D. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic if and only if it can be expanded in a power series in z* in a neighborhood of each point in its domain. If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.

Arithmetic function

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Arithmetic functionIn number theory, an arithmetic (or arithmetical) function is a real or complex valued function (n) defined on the set of natural numbers (i.e. positive integers) that "expresses some arithmetical property of n."[1] An example of an arithmetic function is the non-principal character (mod 4) defined by where is the Kronecker symbol.

To emphasize that they are being thought of as functions rather than sequences, values of an arithmetic function are usually denoted by a(n) rather than an. There is a larger class of number-theoretic functions that do not fit the above definition, e.g. the prime-counting functions. This article provides links to functions of both classes.

Notation and mean that the sum or product is over all prime numbers: Similarly, and mean that the sum or product is over all prime powers with strictly positive

exponent (so 1 is not counted):

and n = 12,

mean that the sum or product is over all positive divisors of n, including 1 and n. E.g., if

The notations can be combined: divisors of n. E.g., if n = 18,

and

mean that the sum or product is over all prime

and similarly E.g., if n = 24,

and

mean that the sum or product is over all prime powers dividing n.

Arithmetic function

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Multiplicative and additive functionsAn arithmetic function a is completely additive if a(mn) = a(m) + a(n) for all natural numbers m and n; completely multiplicative if a(mn) = a(m)a(n) for all natural numbers m and n; Two whole numbers m and n are called coprime if their greatest common divisor is 1; i.e., if there is no prime number that divides both of them. Then an arithmetic function a is additive if a(mn) = a(m) + a(n) for all coprime natural numbers m and n; multiplicative if a(mn) = a(m)a(n) for all coprime natural numbers m and n.

(n), (n), p(n) - prime power decompositionThe fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p1 < p2 < ... < pk are primes and the aj are positive integers. (1 is given by the empty product.) It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent. Define p(n) as the exponent of the highest power of the prime p that divides n. I.e. if p is one of the pi then p(n) = ai, otherwise it is zero. Then

In terms of the above the functions and are defined by (n) = k, (n) = a1 + a2 + ... + ak. To avoid repetition, whenever possible formulas for the functions listed in this article are given in terms of n and the corresponding pi, ai, , and .

Multiplicative functionsk(n), (n), d(n) - divisor sumsk(n) is the sum of the kth powers of the positive divisors of n, including 1 and n, where k is a complex number. 1(n), the sum of the (positive) divisors of n, is usually denoted by (n). Since a positive number to the zero power is one, 0(n) is therefore the number of (positive) divisors of n; it is usually denoted by d(n) or (n) (for the German Teiler = divisors).

Setting k = 0 in the second product gives

Arithmetic function

12

(n) - Euler totient function(n), the Euler totient function, is the number of positive integers not greater than n that are coprime to n.

(n) - Mbius function(n), the Mbius function, is important because of the Mbius inversion formula. See Dirichlet convolution, below.

This implies that (1) = 1. (Because (1) = (1) = 0.)

(n) - Ramanujan tau function(n), the Ramanujan tau function, is defined by its generating function identity:

Although it is hard to say exactly what "arithmetical property of n" it "expresses",[2] ((n) is (2)12 times the nth Fourier coefficient in the q-expansion of the modular discriminant function)[3] it is included among the arithmetical functions because it is multiplicative and it occurs in identities involving certain k(n) and rk(n) functions (because these are also coefficients in the expansion of modular forms).

cq(n) - Ramanujan's sum

cq(n), Ramanujan's sum, is the sum of the nth powers of the primitive qth roots of unity:

Even though it is defined as a sum of complex numbers (irrational for most values of q), it is an integer. For a fixed value of n it is multiplicative in q: If q and r are coprime, Many of the functions mentioned in this article have expansions as series involving these sums; see the article Ramanujan's sum for examples.

Completely multiplicative functions(n) - Liouville function(n), the Liouville function, is defined by

(n) - charactersAll Dirichlet characters (n) are completely multiplicative; e.g. the non-principal character (mod 4) defined in the introduction, or the principal character (mod n) defined by

Arithmetic function

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Additive functions(n) - distinct prime divisors(n), defined above as the number of distinct primes dividing n, is additive

Completely additive functions(n) - prime divisors(n), defined above as the number of prime factors of n counted with multiplicities, is completely additive.

p(n) - prime power dividing nFor a fixed prime p, p(n), defined above as the exponent of the largest power of p dividing n, is completely additive.

Neither multiplicative nor additive(x), (x), (x), (x) - prime count functionsUnlike the other functions listed in this article, these are defined for non-negative real (not just integer) arguments. They are used in the statement and proof of the prime number theorem. (x), the prime counting function, is the number of primes not exceeding x.

A related function counts prime powers with weight 1 for primes, 1/2 for their squares, 1/3 for cubes, ...

(x) and (x), the Chebyshev functions are defined as sums of the natural logarithms of the primes not exceeding x:

(n) - von Mangoldt function(n), the von Mangoldt function, is 0 unless the argument is a prime power, in which case it is the natural log of the prime:

p(n) - partition functionp(n), the partition function, is the number of ways of representing n as a sum of positive integers, where two representations with the same summands in a different order are not counted as being different:

Arithmetic function

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(n) - Carmichael function(n), the Carmichael function, is the smallest positive number such that for all a coprime to n. Equivalently, it is the least common multiple of the orders of the elements of the multiplicative group of integers modulo n. For powers of odd primes and for 2 and 4, (n) is equal to the Euler totient function of n; for powers of 2 greater than 4 it is equal to one half of the Euler totient function of n:

and for general n it is the least common multiple of of each of the prime power factors of n:

h(n) - Class numberh(n), the class number function, is the order of the ideal class group of an algebraic extension of the rationals with discriminant n. The notation is ambiguous, as there are in general many extensions with the same discriminant. See quadratic field and cyclotomic field for classical examples.

rk(n) - Sum of k squaresrk(n) is the number of ways n can be represented as the sum of k squares, where representations that differ only in the order of the summands or in the signs of the square roots are counted as different.

Summation functionsGiven an arithmetic function a(n), its summation function A(x) is defined by

A can be regarded as a function of a real variable. Given a positive integer m, A is constant along open intervals m < x < m + 1, and has a jump discontinuity at each integer for which a(m) 0. Since such functions are often represented by series and integrals, to achieve pointwise convergence it is usual to define the value at the discontinuities as the average of the values to the left and right:

Individual values of arithmetic functions may fluctuate wildly - as in most of the above examples. Summation functions "smooth out" these fluctuations. In some cases it may be possible to find asymptotic behaviour for the summation function for large x. A classical example of this phenomenon[4] is given by d(n), the number of divisors of n:

Arithmetic function

15

Dirichlet convolutionGiven an arithmetic function a(n), let Fa(s), for complex s, be the function defined by the corresponding Dirichlet series (where it converges):[5]

Fa(s) is called a generating function of a(n). The simplest such series, corresponding to the constant function a(n) = 1 for all n, is (s) the Riemann zeta function. The generating function of the Mbius function is the inverse of the zeta function:

Consider two arithmetic functions a and b and their respective generating functions Fa(s) and Fb(s). The product Fa(s)Fb(s) can be computed as follows:

It is a straightforward exercise to show that if c(n) is defined by

then

This function c is called the Dirichlet convolution of a and b, and is denoted by

.

A particularly important case is convolution with the constant function a(n) = 1 for all n, corresponding to multiplying the generating function by the zeta function:

Multiplying by the inverse of the zeta function gives the Mbius inversion formula:

If f is multiplicative, then so is g. If f is completely multiplicative, then g is multiplicative, but may or may not be completely multiplicative. The article multiplicative function has a short proof.

Relations among the functionsThere are a great many formulas connecting arithmetical functions with each other and with the other functions of analysis - in fact, a large part of elementary and analytic number theory is a detailed study of these relations. Here are a few examples:

Dirichlet convolutions where is the Liouville function. [6] [7]

Arithmetic function

16

Mbius inversion [8] [9] Mbius inversion

Mbius inversion

Mbius inversion

where is the Liouville function. [10] Mbius inversion

Sums of squares (Lagrange's four-square theorem). where is the non-principal character (mod 4) defined in the introduction.[11]

where = 2(n). [12][13] [14]

[15] Define the function k*(n) as[16]

That is, if n is odd, k*(n) is the sum of the kth powers of the divisors of n, i.e. k(n), and if n is even it is the sum of the kth powers of the even divisors of n minus the sum of the kth powers of the odd divisors of n. [15] [17] Adopt the convention that Ramanujan's (x) = 0 if x is not an integer.

Arithmetic function

17

[18]

Divisor sum convolutionsHere "convolution" does not mean "Dirichlet convolution" but instead refers to the formula for the coefficients of the product of two power series:

The sequence

is called the convolution or the Cauchy product of the seqences an and bn. See

Eisenstein series for a discussion of the series and functional identities involved in these formulas. [19]

[20]

[20] [21]

[19] [22]

where (n) is Ramanujan's function. [23] [24] Since k(n) (for natural number k) and (n) are integers, the above formulas can be used to prove congruences[25] for the functions. See Tau-function for some examples. Extend the domain of the partition function by setting p(0) = 1. [26] This recurrence can be used to compute p(n).

Prime-count relatedLet be the nth harmonic number. Then is true for every natural number n if and only if the Riemann hypothesis is true. [27]

[28]

Arithmetic function

18

[29] [30] [31]

Miscellaneous and [32] [33] where (n) is Carmichael's function. Further, [34] where (n) is Liouville's function.

[35]

[36]

Note that

[37]

[38] Compare this with 13 + 23 + 33 + ... + n3 = (1 + 2 + 3 + ... + n)2

[39] [40] where (n) is Ramanujan's function. [41]

Arithmetic function

19

Notes[1] [2] [3] [4] [5] Hardy & Wright, intro. to Ch. XVI Hardy, Ramanujan, 10.2 Apostol, Modular Functions ..., 1.15, Ch. 4, and ch. 6 Hardy & Wright, 18.118.2 Hardy & Wright, 17.6, show how the theory of generating functions can be constructed in a purely formal manner with no attention paid to convergence. [6] Hardy & Wright, Thm. 263 [7] Hardy & Wright, Thm. 63 [8] Hardy & Wright, Thm. 288290 [9] Hardy & Wright, Thm. 264 [10] Hardy & Wright, Thm. 296 [11] Hardy & Wright, Thm. 278 [12] Hardy & Wright, Thm. 386 [13] Hardy, Ramanujan, eqs 9.1.2, 9.1.3 [14] Koblitz, Ex. III.5.2 [15] Hardy & Wright, 20.13 [16] Hardy, Ramanujan, 9.7 [17] Hardy, Ramanujan, 9.13 [18] Hardy, Ramanujan, 9.17 [19] Ramanujan, On Certain Arithmetical Functions, Table IV; Papers, p. 146 [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] Koblitz, ex. III.2.8 Koblitz, ex. III.2.3 Koblitz, ex. III.2.2 Koblitz, ex. III.2.4 Apostol, Modular Functions ..., Ex. 6.10 Apostol, Modular Functions..., Ch. 6 Ex. 10 G.H. Hardy, S. Ramannujan, Asymptotic Formul in Combinatory Analysis, 1.3; in Ramannujan, Papers p. 279 See Divisor function. Hardy & Wright, eq. 22.1.2 See prime counting functions. Hardy & Wright, eq. 22.1.1 Hardy & Wright, eq. 22.1.3 Hardy Ramanujan, eq. 3.10.3 Hardy & Wright, 22.13 See Multiplicative group of integers modulo n and Primitive root modulo n. Hardy & Wright, Thm. 329 Hardy & Wright, Thms. 271, 272 Hardy & Wright, eq. 16.3.1 Ramanujan, Some Formul in the Analytic Theory of Numbers, eq. (C); Papers p.133 Ramanujan, Some Formul in the Analytic Theory of Numbers, eq. (F); Papers p.134 Apostol, Modular Functions ..., ch. 6 eq. 4 Apostol, Modular Functions ..., ch. 6 eq. 3

References Tom M. Apostol (1976), Introduction to Analytic Number Theory, Springer Undergraduate Texts in Mathematics, ISBN0387901639 Apostol, Tom M. (1989), Modular Functions and Dirichlet Series in Number Theory (2nd Edition), New York: Springer, ISBN0-387-97127 Hardy, G. H. (1999), Ramanujan: Twelve Lectures on Subjects Suggested by his Life and work, Providence RI: AMS / Chelsea, ISBN978-0821820230 Hardy, G. H.; Wright, E. M. (1980), An Introduction to the Theory of Numbers (Fifth edition), Oxford: Oxford University Press, ISBN978-0198531715 G. J. O. Jameson (2003), The Prime Number Theorem, Cambridge University Press, ISBN0-521-89110-8

Arithmetic function Koblitz, Neal (1984), Introduction to Elliptic Curves and Modular Forms, New York: Springer, ISBN0-387-97966-2 William J. LeVeque (1996), Fundamentals of Number Theory, Courier Dover Publications, ISBN0486689069 Elliott Mendelson (1987), Introduction to Mathematical Logic, CRC Press, ISBN0412808307 Ramanujan, Srinivasa (2000), Collected Papers, Providence RI: AMS / Chelsea, ISBN978-0821820766

20

External links Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions (http://mathstat.carleton.ca/ ~williams/papers/pdf/249.pdf) PDF of a paper by Huard, Ou, Spearman, and Williams. Contains elementary (i.e. not relying on the theory of modular forms) proofs of divisor sum convolutions, formulas for the number of ways of representing a number as a sum of triangular numbers, and related results.

BijectionIn mathematics, a bijection, or a bijective function, is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f(x) = y. It follows from this definition that no unmapped element exists in either X or Y. Alternatively, f is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective). For example, consider the function succ, defined from the set of integers to , that to each integer x associates the integer succ(x) = x + 1. For another example, consider the function sumdif that to each pair (x,y) of real numbers associates the pair sumdif(x,y) = (x+y, xy). A bijective function from a set to itself is also called a permutation.A bijective function, f:XY, where set X is {1,2,3,4} and set Y is {A,B,C,D}. For example, f(1)=D.

The set of all bijections from X to Y is denoted as X Y. (Sometimes this notation is reserved for binary relations, and bijections are denoted by X Y instead.) Occasionally, the set of permutations of a single set X may be denoted X!. Bijective functions play a fundamental role in many areas of mathematics, for instance in the definition of isomorphism (and related concepts such as homeomorphism and diffeomorphism), permutation group, projective map, and many others.

Bijection

21

Composition and inversesA function f is bijective if and only if its inverse relation f 1 is a function. In that case, f 1 is also a bijection. The composition of two bijections . Conversely, if the composition of two functions is bijective, we can only say that f is injective and g is surjective. A relation f from X to Y is a bijective function if and only if there exists another relation g from Y to X such that is the identity function on X, and is the identity function on Y. Consequently, the sets have the same cardinality. and is a bijection. The inverse of is

Bijections and cardinalityIf X and Y are finite sets, then there exists a A bijection composed of an injection (left) and a surjection (right). bijection between the two sets X and Y iff X and Y have the same number of elements. Indeed, in axiomatic set theory, this is taken as the very definition of "same number of elements", and generalising this definition to infinite sets leads to the concept of cardinal number, a way to distinguish the various sizes of infinite sets.

Examples and counterexamples For any set X, the identity function idX from X to X, defined by idX(x) = x, is bijective. The function f from the real line R to R defined by f(x) = 2x + 1 is bijective, since for each y there is a unique x = (y1)/2 such that f(x) = y. The exponential function g:R R, with g(x) = ex, is not bijective: for instance, there is no x in R such that g(x) = 1, showing that g is not surjective. However if the codomain is restricted to the positive real numbers R+ = (0,+), then g becomes bijective; its inverse is the natural logarithm function ln. The function h:R [0,+) with h(x) = x is not bijective: for instance, h(1) = h(+1) = 1, showing that h is not injective. However, if the domain is restricted to [0,+), then h becomes bijective; its inverse is the positive square root function.

Properties A function f from the real line R to R is bijective if and only if its plot is intersected by any horizontal or vertical line at exactly one point. If X is a set, then the bijective functions from X to itself, together with the operation of functional composition (), form a group, the symmetric group of X, which is denoted variously by S(X), SX, or X! (the last reads "X factorial"). For a subset A of the domain with cardinality |A| and subset B of the codomain with cardinality |B|, one has the following equalities: |f(A)| = |A| and |f1(B)| = |B|. If X and Y are finite sets with the same cardinality, and f:XY, then the following are equivalent: 1. f is a bijection.

Bijection 2. f is a surjection. 3. f is an injection. At least for a finite set S, there is a bijection between the set of possible total orderings of the elements and the set of bijections from S to S. That is to say, the number of permutations of elements of S is the same as the number of total orderings of that setnamely, n!.

22

Bijections and category theoryFormally, bijections are precisely the isomorphisms in the category Set of sets and functions. However, the bijections are not always the isomorphisms. For example, in the category Top of topological spaces and continuous functions, the isomorphisms must be homeomorphisms in addition to being bijections.

See also Category theory Injective function Symmetric group Surjective function

Bijective numeration Bijective proof Cardinality

ReferencesWeisstein, Eric W., "Bijection [1]" from MathWorld.

External links Earliest Uses of Some of the Words of Mathematics: entry on Injection, Surjection and Bijection has the history of Injection and related terms. [2]

References[1] http:/ / mathworld. wolfram. com/ Bijection. html [2] http:/ / jeff560. tripod. com/ i. html

Binary function

23

Binary functionIn mathematics, a binary function, or function of two variables, is a function which takes two inputs. Precisely stated, a function is binary if there exists sets such that

where

is the Cartesian product of

and

For example, if Z is the set of integers, N+ is the set of natural numbers (except for zero), and Q is the set of rational numbers, then division is a binary function from Z and N+ to Q. Set-theoretically, one may represent a binary function as a subset of the Cartesian product X Y Z, where (x,y,z) belongs to the subset if and only if f(x,y) = z. Conversely, a subset R defines a binary function if and only if, for any x in X and y in Y, there exists a unique z in Z such that (x,y,z) belongs to R. We then define f (x,y) to be this z. Alternatively, a binary function may be interpreted as simply a function from X Y to Z. Even when thought of this way, however, one generally writes f (x,y) instead of f((x,y)). (That is, the same pair of parentheses is used to indicate both function application and the formation of an ordered pair.) In turn, one can also derive ordinary functions of one variable from a binary function. Given any element x of X, there is a function f x, or f (x,), from Y to Z, given by f x(y) := f (x,y). Similarly, given any element y of Y, there is a function f y, or f (,y), from X to Z, given by f y(x) := f (x,y). (In computer science, this identification between a function from X Y to Z and a function from X to ZY is called Currying.) The various concepts relating to functions can also be generalised to binary functions. For example, the division example above is surjective (or onto) because every rational number may be expressed as a quotient of an integer and a natural number. This example is injective in each input separately, because the functions f x and f y are always injective. However, it's not injective in both variables simultaneously, because (for example) f (2,4) = f (1,2). One can also consider partial binary functions, which may be defined only for certain values of the inputs. For example, the division example above may also be interpreted as a partial binary function from Z and N to Q, where N is the set of all natural numbers, including zero. But this function is undefined when the second input is zero. A binary operation is a binary function where the sets X, Y, and Z are all equal; binary operations are often used to define algebraic structures. In linear algebra, a bilinear transformation is a binary function where the sets X, Y, and Z are all vector spaces and the derived functions f x and fy are all linear transformations. A bilinear transformation, like any binary function, can be interpreted as a function from X Y to Z, but this function in general won't be linear. However, the bilinear transformation can also be interpreted as a single linear transformation from the tensor product X Y to Z. The concept of binary function generalises to ternary (or 3-ary) function, quaternary (or 4-ary) function, or more generally to n-ary function for any natural number n. A 0-ary function to Z is simply given by an element of Z. One can also define an A-ary function where A is any set; there is one input for each element of A. In category theory, n-ary functions generalise to n-ary morphisms in a multicategory. The interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category. The construction of the derived morphisms of one variable will work in a closed monoidal category. The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.

Bochner measurable function

24

Bochner measurable functionIn mathematics specifically, in functional analysis a Bochner-measurable function taking values in a Banach space is a function that equals a.e. the limit of a sequence of measurable countably-valued functions, i.e.,

where the functions

each have a countable range and for which the pre-image

is measurable for

eachx. The concept is named after Salomon Bochner. Bochner-measurable functions are sometimes called strongly measurable,

-measurable or just measurable (or

uniformly measurable in case that the Banach space is the space of continuous linear operators between Banach spaces).

PropertiesThe relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem. Function f is almost surely separably valued (or essentially separably valued) if there exists a subset NX with (N)=0 such that f(X\N)B is separable. A function :XB defined on a measure space (X,,) and taking values in a Banach space B is (strongly) measurable (with respect to and the Borel -algebra on B) if and only if it is both weakly measurable and almost surely separably valued. In the case that B is separable, since any subset of a separable Banach space is itself separable, one can take N above to be empty, and it follows that the notions of weak and strong measurability agree when B is separable.

References Showalter, Ralph E. (1997). "Theorem III.1.1". Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p.103. MR1422252. ISBN0-8218-0500-2..

Bounded function

25

Bounded functionIn mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M < such that

for all x in X. Sometimes, if for all x in X, then the function is said to be for all x in X,

bounded above by A. On the other hand, if

then the function is said to be bounded below by B. The concept should not be confused with that of a bounded operator. An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = ( a0, a1, a2, ... ) is bounded if there exists a real number M < such that |an| M for every natural number n. The set of all bounded sequences, equipped with a vector space structure, forms a sequence space.A schematic illustration of a bounded function (red) and an unbounded one (blue). Intuitively, the graph of a bounded function stays within a horizontal band, while the graph of an unbounded function does not.

This definition can be extended to functions taking values in a metric space Y. Such a function f defined on some set X is called bounded if for some a in Y there exists a real number M < such that

for all x in X. If this is the case, there is also such an M for each other a.

Examples The function f:R R defined by f(x)=sin x is bounded. The sine function is no longer bounded if it is defined over the set of all complex numbers The function

defined for all real x which do not equal 1 or 1 is not bounded. As x gets closer to 1 or to 1, the values of this function get larger and larger in magnitude. This function can be made bounded if one considers its domain to be, for example, [2, ). The function

defined for all real x is bounded. Every continuous function f:[0,1] R is bounded. This is really a special case of a more general fact: Every continuous function from a compact space into a metric space is bounded. The function f which takes the value 0 for x rational number and 1 for x irrational number is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on [0,1] is much bigger than the set of continuous functions on that interval.

Cauchy-continuous function

26

Cauchy-continuous functionIn mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful property that they can always be (uniquely) extended to the Cauchy completion of their domain.

DefinitionLet X and Y be metric spaces, and let f be a function from X to Y. Then f is Cauchy-continuous if and only if, given any Cauchy sequence (x1, x2, ) in X, the sequence (f(x1), f(x2), ) is a Cauchy sequence in Y.

PropertiesEvery uniformly continuous function is also Cauchy-continuous, and any Cauchy-continuous function is continuous. Conversely, if X is a complete space, then every continuous function on X is Cauchy-continuous too. More generally, even if X is not complete, as long as Y is complete, then any Cauchy-continuous function from X to Y can be extended to a function defined on the Cauchy completion of X; this extension is necessarily unique.

Examples and non-examplesSince the real line is complete, the Cauchy-continuous functions on are the same as the continuous ones. On the subspace of rational numbers, however, matters are different. For example, define a two-valued function so that f(x) is 0 when x2 is less than 2 but 1 when x2 is greater than 2. (Note that x2 is never equal to 2 for any rational number x.) This function is continuous on but not Cauchy-continuous, since it can't be extended to . On the other hand, any uniformly continuous function on must be Cauchy-continuous. For a non-uniform example on , let f(x) be 2x; this is not uniformly continuous (on all of ), but it is Cauchy-continuous. A Cauchy sequence (y1, y2, ) in Y can be identified with a Cauchy-continuous function from {1, 1/2, 1/3, } to Y, defined by f(1/n)= yn. If Y is complete, then this can be extended to {1, 1/2, 1/3, , 0}; f(0) will be the limit of the Cauchy sequence.

GeneralisationsCauchy continuity makes sense in situations more general than metric spaces, but then one must move from sequences to nets (or equivalently filters). The definition above applies, as long as the Cauchy sequence (x1, x2, ) is replaced with an arbitrary Cauchy net. Equivalently, a function f is Cauchy-continuous if and only if, given any Cauchy filter F on X, then f(F) is a Cauchy filter on Y. This definition agrees with the above on metric spaces, but it also works for uniform spaces and, most generally, for Cauchy spaces. Any directed set A may be made into a Cauchy space. Then given any space Y, the Cauchy nets in Y indexed by A are the same as the Cauchy-continuous functions from A to Y. If Y is complete, then the extension of the function to A {} will give the value of the limit of the net. (This generalises the example of sequences above, where 0 is to be interpreted as 1/.)

Cauchy-continuous function

27

References Eva Lowen-Colebunders (1989). Function Classes of Cauchy Continuous Maps. Dekker, New York.

Closed convex functionIn mathematics, a convex function is called closed if its epigraph is a closed set.

PropertiesA closed convex function f is the pointwise supremum of the collection of all affine functions h such that hf.

References Rockafellar, R. Tyrell, Convex Analysis, Princeton University Press (1996). ISBN 0-691-01586-4

Coarse functionIn mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous.[1] Although continuous functions are usually observed on a small scale, coarse functions are usually observed on a large scale.[1] [2]

See also Continuous function

References[1] Chul-Woo Lee and Jared Duke (2007), Coarse Function Value Theorems (http:/ / www. rose-hulman. edu/ mathjournal/ archives/ 2007/ vol8-n2/ paper4/ v8n2-4pd. pdf). Rose-Hulman Undergraduate Mathematics Journal 8 (2) [2] Dongarra, Jack; Madsen, Kaj; Wasniewski, Jerzy, eds (2006). Applied Parallel Computing: State of the Art in Scientific Computing (http:/ / books. google. com/ books?id=ZqFkI1MufzMC). Germany: Springer-Verlag Berlin Heidelberg. pp.316322. ISBN978-3-540-29067-4. .

Completely multiplicative function

28

Completely multiplicative functionIn number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. Especially in number theory, a weaker condition is also important, respecting only products of coprime numbers, and such functions are called multiplicative functions. Outside of number theory, the term "multiplicative function" is often taken to be synonymous with "completely multiplicative function" as defined in this article.

DefinitionA completely multiplicative function (or totally multiplicative function) is an arithmetic function (that is, a function whose domain is the natural numbers), such that f(1) = 1 and f(ab) = f(a) f(b) holds for all positive integers a and b. Without the requirement that f(1) = 1, one could still have f(1) = 0, but then f(a) = 0 for all positive integers a, so this is not a very strong restriction.

ExamplesThe easiest example of a multiplicative function is a monomial: For any particular positive integer n, define f(a) = an.

PropertiesA completely multiplicative function is completely determined by its values at the prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if n is a product of powers of distinct primes, say n = pa qb ..., then f(n) = f(p)a f(q)b ...

Concave function

29

Concave functionIn mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex cap or upper convex.

DefinitionA real-valued function f defined on an interval (or on any convex set C of some vector space) is called concave if, for any two points x and y in its domain C and any t in [0,1], we have

A function is called strictly concave if

for any t in (0,1) and x y. For a function f:RR, this definition merely states that for every z between x and y, the point (z, f(z) ) on the graph of f is above the straight line joining the points (x, f(x) ) and (y, f(y) ).

A function f(x) is a quasiconcave if the upper contour sets of the function sets.[1]

are convex

PropertiesA function f(x) is concave over a convex set if and only if the function f(x) is a convex function over the set. A differentiable function f is concave on an interval if its derivative function f is monotonically decreasing on that interval: a concave function has a decreasing slope. ("Decreasing" here means "non-increasing", rather than "strictly decreasing", and thus allows zero slopes.) For a twice-differentiable function f, if the second derivative, f (x), is positive (or, if the acceleration is positive), then the graph is convex; if f (x) is negative, then the graph is concave. Points where concavity changes are inflection points. If a convex (i.e., concave upward) function has a "bottom", any point at the bottom is a minimal extremum. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a maximal extremum.

Concave function If f(x) is twice-differentiable, then f(x) is concave if and only if f (x) is non-positive. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x4. If f is concave and differentiable then[2]

30

A continuous function on C is concave if and only if for any x and y in C

If a function f is concave, and f(0) 0, then f is subadditive. Proof: since f is concave, let y = 0,

Examples The functions Any linear function The function and are concave, as the second derivative is always negative. is both concave and convex. is concave on the interval .

The function , where is the determinant of matrix nonnegative-definite matrix B, is concave[3] . Practical application: rays bending in Computation of radiowave attenuation in the atmosphere.

References[1] Varian, Hal A. (1992) Microeconomic Analysis. Third Edition. W.W. Norton and Company. p. 496 [2] Varian, Hal A. (1992) Microeconomic Analysis. Third Edition. W.W. Norton and Company. p. 489 [3] Thomas M. Cover and J. A. Thomas (1988). "Determinant inequalities via information theory". SIAM journal on matrix analysis and applications 9 (3): 384392.

Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. p.779. ISBN0470183527.

Constant function

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Constant functionIn mathematics, a constant function is a function whose values do not vary and thus are constant. For example, if we have the function f(x) = 4, then f is constant since f maps any value to 4. More formally, a function f:A B is a constant function if f(x) = f(y) for all x and y in A. Every empty function is constant, vacuously, since there are no x and y in A for which f(x) and f(y) are different when A is the empty set. Some find it more convenient, however, to define constant function so as to exclude empty functions. In the context of polynomial functions, a non-zero constant function is called a polynomial of degree zero.

PropertiesConstant functions can be characterized with respect to function composition in two ways. The following are equivalent: 1. f:A B is a constant function. 2. For all functions g, h:C A, f o g = f o h, (where "o" denotes function composition). 3. The composition of f with any other function is also a constant function. The first characterization of constant functions given above, is taken as the motivating and defining property for the more general notion of constant morphism in category theory. In contexts where it is defined, the derivative of a function measures how that function varies with respect to the variation of some argument. It follows that, since a constant function does not vary, its derivative, where defined, will be zero. Thus for example: If f is a real-valued function of a real variable, defined on some interval, then f is constant if and only if the derivative of f is everywhere zero. For functions between preordered sets, constant functions are both order-preserving and order-reversing; conversely, if f is both order-preserving and order-reversing, and if the domain of f is a lattice, then f must be constant. Other properties of constant functions include: Every constant function whose domain and codomain are the same is idempotent. Every constant function between topological spaces is continuous. A function on a connected set is locally constant if and only if it is constant.

References Herrlich, Horst and Strecker, George E., Category Theory, Allen and Bacon, Inc. Boston (1973) Constant function [1] on PlanetMath

References[1] http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=4727

Continuous function

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Continuous functionIn mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article. As an example, consider the function h(t) which describes the height of a growing flower at time t. This function is continuous. In fact, there is a dictum of classical physics which states that in nature everything is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous. (However, if one assumes a discrete set as the domain of function M, for instance the set of points of time at 4:00 PM on business days, then M becomes continuous function, as every function whose domain is a discrete subset of reals is.)

Real-valued continuous functionsHistorical infinitesimal definitionCauchy defined continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34).

Definition in terms of limitsSuppose we have a function that maps real numbers to real numbers and whose domain is some interval, like the functions h and M above. Such a function can be represented by a graph in the Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps". In general, we say that the function f is continuous at some point c of its domain if, and only if, the following holds: The limit of f(x) as x approaches c through domain of f does exist and is equal to f(c); in mathematical notation, . If the point c in the domain of f is not a limit point of the domain, then this condition is vacuously true, since x cannot approach c through values not equal c. Thus, for example, every function whose domain is the set of all integers is continuous. We call a function continuous if and only if it is continuous at every point of its domain. More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset. The notation C() or C0() is sometimes used to denote the set of all continuous functions with domain . Similarly, C1() is used to denote the set of differentiable functions whose derivative is continuous, C() for the twice-differentiable functions whose second derivative is continuous, and so on (see differentiability class). In the field of computer graphics, these three levels are sometimes called g0 (continuity of position), g1 (continuity of tangency), and g2 (continuity of curvature). The notation C(n, )() occurs in the definition of a more subtle concept, that of Hlder continuity.

Continuous function

33

Weierstrass definition (epsilon-delta) of continuous functionsWithout resorting to limits, one can define continuity of real functions as follows. Again consider a function that maps a set of real numbers to another set of real numbers, and suppose c is an element of the domain of . The function is said to be continuous at the point c if the following holds: For any number >0, however small, there exists some number >0 such that for all x in the domain of with c0 such that for all xI,:

A form of this epsilon-delta definition of continuity was first given by Bernard Bolzano in 1817. Preliminary forms of a related definition of the limit were given by Cauchy,[1] though the formal definition and the distinction between pointwise continuity and uniform continuity were first given by Karl Weierstrass. More intuitively, we can say that if we want to get all the (x) values to stay in some small neighborhood around (c), we simply need to choose a small enough neighborhood for the x values around c, and we can do that no matter how small the (x) neighborhood is; is then continuous atc. In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.

Definition using oscillation

The failure of a function to be continuous at a point is quantified by its oscillation.

Continuity can also be defined in terms of oscillation: a function is continuous at a point x0 if and only if the oscillation is zero;[2] in symbols, A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point. This definition is useful in descriptive set theory to study the set of discontinuities and continuous points the continuous points are the intersection of the sets where the oscillation is less than (hence a G set) and gives a very quick proof of one direction of the Lebesgue integrability condition.[3] The oscillation is equivalent to the - definition by a simple re-arrangement, and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given 0 there is no that satisfies the - definition, then the oscillation is at least 0, and conversely if for every there is a desired , the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.

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Definition using the hyperrealsNon-standard analysis is a way of making Newton-Leibniz-style infinitesimals mathematically rigorous. The real line is augmented by the addition of infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows. A function from the reals to the reals is continuous if its natural extension to the hyperreals has the property that for real x and infinitesimal dx, (x+dx) (x) is infinitesimal.[4] In other words, an infinitesimal increment of the independent variable corresponds to an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.

Examples All polynomial functions are continuous. If a function has a domain which is not an interval, the notion of a continuous function as one whose graph you can draw without taking your pencil off the paper is not quite correct. Consider the functions f(x) = 1/x and g(x) = (sinx)/x. Neither function is defined at x = 0, so each has domain R \ {0} of real numbers except 0, and each function is continuous. The question of continuity at x = 0 does not arise, since x = 0 is neither in the domain of f nor in the domain of g. The function f cannot be extended to a continuous function whose domain is R, since no matter what value is assigned at 0, the resulting function will not be continuous. On the other hand, since the limit of g at 0 is 1, g can be extended continuously to R by defining its value at 0 to be 1. The exponential functions, logarithms, square root function, trigonometric functions and absolute value function are continuous. Rational functions, however, are not necessarily continuous on all of R. An example of a rational continuous function is f(x)=1x-2. The question of continuity at x= 2 does not arise, since x = 2 is not in the domain of f. An example of a discontinuous function is the function f defined by f(x) = 1 if x > 0, f(x) = 0 if x 0. Pick for instance = 12. There is no -neighborhood around x = 0 that will force all the f(x) values to be within of f(0). Intuitively we can think of this type of discontinuity as a sudden jump in function values. Another example of a discontinuous function is the signum or sign function. A more complicated example of a discontinuous function is Thomae's function. Dirichlet's function

is nowhere continuous.

Facts about continuous functionsIf two functions f and g are continuous, then f + g, fg, and f/g are continuous. (Note. The only possible points x of discontinuity of f/g are the solutions of the equation g(x) = 0; but then any such x does not belong to the domain of the function f/g. Hence f/g is continuous on its entire domain, or - in other words - is continuous.) The composition f o g of two continuous functions is continuous. If a function is differentiable at some point c of its domain, then it is also continuous at c. The converse is not true: a function that is continuous at c need not be differentiable there. Consider for instance the absolute value function at c=0.

Continuous function Intermediate value theorem The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states: If the real-valued function f is continuous on the closed interval [a,b] and k is some number between f(a) and f(b), then there is some number c in [a,b] such that f(c)=k. For example, if a child grows from 1m to 1.5m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25m. As a consequence, if f is continuous on [a,b] and f(a) and f(b) differ in sign, then, at some point c in [a,b], f(c) must equal zero. Extreme value theorem The extreme value theorem states that if a function f is defined on a closed interval [a,b] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists c[a,b] with f(c) f(x) for all x[a,b]. The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval (a,b) (or any set that is not both closed and bounded), as, for example, the continuous function f(x) = 1/x, defined on the open interval (0,1), does not attain a maximum, being unbounded above.

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Directional continuity

A right continuous function

A left continuous function

A function may happen to be continuous in only one direction, either from the "left" or from the "right". A right-continuous function is a function which is continuous at all points when approached from the right. Technically, the formal definition is similar to the definition above for a continuous function but modified as follows: The function is said to be right-continuous at the point c if the following holds: For any number > 0 however small, there exists some number > 0 such that for all x in the domain with c < x < c + , the value of (x) will satisfy

Notice that x must be larger than c, that is on the right of c. If x were also allowed to take values less thanc, this would be the definition of continuity. This restriction makes it possible for the function to have a discontinuity atc, but still be right continuous atc, as pictured. Likewise a left-continuous function is a function which is continuous at all points when approached from the left, that is, c < x < c.

Continuous function A function is continuous if and only if it is both right-continuous and left-continuous.

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Continuous functions between metric spacesNow consider a function f from one metric space (X, dX) to another metric space (Y, dY). Then f is continuous at the point c in X if for any positive real number , there exists a positive real number such that all x in X satisfying dX(x, c) < will also satisfy dY(f(x), f(c)) < . This can also be formulated in terms of sequences and limits: the function f is continuous at the point c if for every sequence (xn) in X with limit lim xn = c, we have lim f(xn) = f(c). Continuous functions transform limits into limits. This latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence (xn) in X with limit c, the sequence (f(xn)) is a Cauchy sequence, and c is in the domain of f. Continuous functions transform convergent sequences into Cauchy sequences. The set of points at which a function between metric spaces is continuous is a G set this follows from the - definition of continuity.

Continuous functions between topological spacesThe above definitions of continuous functions can be generalized to functions from one topological space to another in a natural way; a function f : X Y, where X and Y are topological spaces, is continuous if and only if for every open set V Y, the inverse image

Continuity of a function at a point

is open. However, this definition is often difficult to use directly. Instead, suppose we have a function f from X to Y, where X, Y are topological spaces. We say f is continuous at x for some xX if for any neighborhood V of f(x), there is a neighborhood U of x such that f(U)V. Although this definition appears complex, the intuition is that no matter how "small" V becomes, we can always find a U containing x that will map inside it. If f is continuous at every xX, then we simply say f is continuous. In a metric space, it is equivalent to consider the neighbourhood system of open balls centered at x and f(x) instead of all neighborhoods. This leads to the standard - definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric space, however, which has a notion of distance. Note, however, that if the target space is Hausdorff, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous.

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DefinitionsSeveral equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function. Open and closed set definition The most common notion of continuity in topology defines continuous functions as those functions for which the preimages (or inverse images) of open sets are open. Similar to the open set formulation is the closed set formulation, which says that preimages (or inverse images) of closed sets are closed. Neighborhood definition Definitions based on preimages are often difficult to use directly. Instead, suppose we have a function f : X Y, where X and Y are topological spaces.[5] We say f is continuous at x for some xX if for any neighborhood V of f(x), there is a neighborhood U of x such that f(U)V. Although this definition appears complicated, the intuition is that no matter how "small" V becomes, we can always find a U containing x that will map inside it. If f is continuous at every xX, then we simply say f is continuous.

In a metric space, it is equivalent to consider the neighbourhood system of open balls centered at x and f(x) instead of all neighborhoods. This leads to the standard - definition of a continuous function from real analysis, which says roughly that a function is continuous if all points close to x map to points close to f(x). This only really makes sense in a metric space, however, which has a notion of distance. Note, however, that if the target space is Hausdorff, it is still true that f is continuous at a if and only if the limit of f as x approaches a is f(a). At an isolated point, every function is continuous. Sequences and nets In several contexts, the topology of a space is conveniently specified in terms of limit points. In many instances, this is accomplished by specifying when a point is the limit of a sequence, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a directed set, known as nets. A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition. In detail, a function f : X Y is sequentially continuous if whenever a sequence (xn) in X converges to a limit x, the sequence (f(xn)) converges to f(x). Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If X is a first-countable space, then the converse also holds: any function preserving sequential limits is continuous. In particular, if X is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called sequential spaces.) This motivates the

Continuous function consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions. Closure operator definition Given two topological spaces (X,cl) and (X' ,cl') where cl and cl' are two closure operators then a function

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is continuous if for all subsets A of X

One might therefore suspect that given two topological spaces (X,int) and (X' ,int') where int and int' are two interior operators then a function

is continuous if for all subsets A of X

or perhaps if

however, neither of these conditions is either necessary or sufficient for continuity. Instead, we must resort to inverse images: given two topological spaces (X,int) and (X' ,int') where int and int' are two interior operators then a function

is continuous if for all subsets A of X '

We can also write that given two topological spaces (X,cl) and (X' ,cl') where cl and cl' are two closure operators then a function

is continuous if for all subsets A of X '

Closeness relation definition Given two topological spaces (X,) and (X' ,') where and ' are two closeness relations then a function

is continuous if for all points x and of X and all subsets A of X,

This is another way of writing the closure operator definition.

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Useful properties of continuous mapsSome facts about continuous maps between topological spaces: If f : X Y and g : Y Z are continuous, then so is the composition g f : X Z. If f : X Y is continuous and X is compact, then f(X) is compact. X is connected, then f(X) is connected. X is path-connected, then f(X) is path-connected. X is Lindelf, then f(X) is Lindelf. X is separable, then f(X) is separable. The identity map idX : (X, 2) (X, 1) is continuous if and only if 1 2 (see also comparison of topologies).

Other notesIf a set is given the discrete topology, all functions with that space as a domain are continuous. If the domain set is given the indiscrete topology and the range set is at least T0, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous. Given a set X, a partial ordering can be defined on the possible topologies on X. A continuous function between two topological spaces stays continuous if we strengthen the topology of the domain space or weaken the topology of the codomain space. Thus we can consider the continuity of a given function a topological property, depending only on the topologies of its domain and codomain spaces. For a function f from a topological space X to a set S, one defines the final topology on S by letting the open sets of S be those subsets A of S for which f1(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is coarser than the final topology on S. Thus the final topology can be characterized as the finest topology on S which makes f continuous. If f is surjective, this topology is canonically identified with the quotient topology under the equivalence relation defined by f. This construction can be generalized to an arbitrary family of functions X S. Dually, for a function f from a set S to a topological space, one defines the initial topology on S by letting the open sets of S be those subsets A of S for which f(A) is open in X. If S has an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus the initial topology can be characterized as the coarsest topology on S which makes f continuous. If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X. This construction can be generalized to an arbitrary family of functions S X. Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. In fact, if an open map f has an inverse, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. If a function is a bijection, then it has an inverse function. The inverse of a continuous bijection is open, but need not be continuous. If it is, this special function is called a homeomorphism. If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is automatically a homeomorphism.

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Continuous functions between partially ordered setsIn order theory, continuity of a function between posets is Scott continuity. Let X be a complete lattice, then a function f : X X is continuous if, for each subset Y of X, we have supf(Y) = f(supY).

Continuous binary relationA binary relation R on A is continuous if R(a, b) whenever there are sequences (ak)i and (bk)i in A which converge to a and b respectively for which R(ak,bk) for all k. Clearly, if one treats R as a characteristic function in two variables, this definition of continuous is identical to that for continuous functions.

Continuity spaceA continuity space[6] [7] is a generalization of metric spaces and posets, which uses the concept of quantales, and that can be used to unify the notions of metric spaces and domains.[8]

See also Absolute continuity Bounded linear operator Classification of discontinuities Coarse function Continuous functor Continuous stochastic process Dini continuity Discrete function Equicontinuity Lipschitz continuity Normal function Piecewise Scott continuity Semicontinuity Smooth function Symmetrically continuous function Uniform continuity

Notes[1] Grabiner, Judith V. (March 1983). "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus" (http:/ / www. maa. org/ pubs/ Calc_articles/ ma002. pdf). The American Mathematical Monthly 90 (3): 185194. doi:10.2307/2975545. . [2] Introduction to Real Analysis (http:/ / ramanujan. math. trinity. edu/ wtrench/ texts/ TRENCH_REAL_ANALYSIS. PDF), updated April 2010, William F. Trench, Theorem 3.5.2, p. 172 [3] Introduction to Real Analysis (http:/ / ramanujan. math. trinity. edu/ wtrench/ texts/ TRENCH_REAL_ANALYSIS. PDF), updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171177 [4] http:/ / www. math. wisc. edu/ ~keisler/ calc. html [5] f is a function f : X Y between two topological spaces (X,TX) and (Y,TY). That is, the function f is defined on the elements of the set X, not on the elements of the topology TX. However continuity of the function does depend on the topologies used. [6] Quantales and continuity spaces (http:/ / citeseerx. ist. psu. edu/ viewdoc/ download?doi=10. 1. 1. 48. 851& rep=rep1& type=pdf), RC Flagg Algebra Universalis, 1997 [7] All topologies come from generalized metrics, R Kopperman - American Mathematical Monthly, 1988 [8] Continuity spaces: Reconciling domains and metric spaces, B Flagg, R Kopperman - Theoretical Computer Science, 1997

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References Visual Calculus (http://archives.math.utk.edu/visual.calculus/) by Lawrence S. Husch, University of Tennessee (2001)

Convex functionIn mathematics, a real-valued function defined on an interval (or on any convex subset of some vector space) is called convex, concave upwards, concave up or convex cup, if for any two points and in its domain X and any ,

Convex function on an interval.

A function (in black) is convex if and only if the region above its graph (in green) is a convex set.

A function is called strictly convex if

for every

,

, and

. may not lie in the

Note that the function must be defined over a convex set, otherwise the point function domain. A function is said to be (strictly) concave if is (strictly) convex.

Pictorially, a function is called 'convex' if the function lies below or on the straight line segment connecting two points, for any two points in the interval. Sometimes an alternative definition is used:

Convex function A function is convex if its epigraph (the set of points lying on or above the graph) is a convex set. These two definitions are equivalent, i.e., one holds if and only if the other one is true.

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PropertiesSuppose is a function of one real variable defined on an interval, and let

(note that R(x,y) is the slope of the red line in the above drawing; note also that the function R is symmetric in x,y). is convex if and only if R(x,y) is monotonically non-decreasing in x, for y fixed (or viceversa). This characterization of convexity is quite useful to prove the following results. A convex function defined on some open interval C is continuous on C and Lipschitz continuous on any closed subinterval. admits left and right derivatives, and these are monotonically non-decreasing. As a consequence, is differentiable at all but at most countably many points. If C is closed, then may fail to be continuous at the endpoints ofC (an example is shown in the examples' section). A function is midpoint convex on an interval C if

for all x and y inC. This condition is only slightly weaker than convexity. For example, a real valued Lebesgue measurable function that is midpoint convex will be convex.[1] In particular, a continuous function that is midpoint convex will be convex. A differentiable function of one variable is convex on an interval if and only if its derivative is monotonically non-decreasing on that interval. If a function is differentiable and convex then it is also continuously differentiable. A continuously differentiable functi