ANALYSIS - Knaust

ANALYSIS

Fall 2004

Helmut KnaustDepartment of Mathematical SciencesThe University of Texas at El PasoEl Paso TX 79968-0514

ROBERT L. M OORE

1882–1974

Preface

The Moore Method. This course uses the “Moore Method” named afterthe eminent mathematician Robert L. Moore, who taught at the Uni-versity of Texas at Austin from 1920–69. The basic idea behind thismethod is that you can only learn how to do mathematics by doingmathematics. Here are two famous quotes attributed to R.L. Moore:

“There is only one math book, and this book has only onepage with a single sentence: Do what you can!’”

“That Student is Taught the Best Who is Told the Least.”

Ground Rules. Expect this course to be quite different from other mathe-matics courses you have taken. Here are the ground rules we will beoperating under:

• These notes contain “exercises” and “tasks”. You will solvethese problems at home and then present the solutions in class. Iwill call on students at random to present “exercises”; I will callon volunteers to present solutions to the “tasks”.

• When you are in the audience, you are expected to be activelyengaged in the presentation. This means checking to see if ev-ery step of the presentation is clear and convincing to you, andspeaking up when it is not. When there are gaps in the reason-ing, the class will work together to fill the gaps. At all times, theconversation will be guided by the principles of “mathematicallyaccountable” talk.

• The instructor will only serve as a moderator. My major contri-bution in class will consist of asking guiding and probing ques-tions. I will also occasionally give short presentations to puttopics into a wider context, or to briefly talk about additionalconcepts not dealt with in the notes.

• You may use only the notes provided and your own class notes;you are not allowed to consult other books or materials. Youmust not talk to people outside of class about the assignments. Ifyou collaborate with other class participants, you must acknowl-edge their contribution during your presentation. Exemptionsfrom these restrictions require prior approval by the instructor.

• An important resource for you is your instructor. I expect fre-quent visits from all of you during my office hours—many morevisits than in a “normal” class. Among other things, you prob-ably will want to come to my office to ask questions about con-cepts and assigned problems, you will probably occasionallywant to show me your work before presenting it in class, andyou probably will have times when you just want to talk aboutthe frustrations you may experience.

• It is of paramount importance that we all agree to create a classatmosphere that is supportive and non-threatening to all partic-ipants. Disparaging remarks will be tolerated neither from stu-dents nor from the instructor.

Historical Perspective. This course gives an “Introduction to Analysis”.After its discovery, Calculus turned out to be extremely useful in solv-ing problems in Physics. Ad hoc justifications were used by the gen-eration of mathematicians following Newton and Leibniz, and evenlater by mathematicians such as Euler, Lagrange and Laplace. Amathematical argument given by Euler, for example, did not differmuch from the kind of “explanations” you have seen in your Calculusclasses.

In the first third of the nineteenth century, with the advent of Fourierseries, fundamental problems with this approach of doing mathemat-ics arose: The leading mathematicians in Europe just did not knowany more what was right and what was wrong! This led to the questfor putting the concepts of Calculus on a sound foundational basis:What exactly does it mean for a sequence to converge? What doesit mean for a function to be differentiable? When is the integral ofan infinite sum of functions equal to the infinite sum of the integralsof the functions? Etc, etc. As these fundamental questions were in-vestigated and consequently answered, the word “Analysis” becamethe customary term to describe this kind of “Rigorous Calculus”. Theprogress in Analysis during the latter part of the nineteenth centuryand the rapid progress in the twentieth century would not have beenpossible without this revitalization of Calculus.

Consequently, this semester we will investigate (or in many cases re-visit) the fundamental concepts in single-variable Calculus: Sequencesand their convergence behavior, local and global consequences of con-

tinuity, properties of differentiable functions, integrability, and the re-lation between differentiability and integrability.

Contents

1 Introduction 1

1.1 The Set of Natural Numbers. . . . . . . . . . . . . . . . . 1

1.2 Integers, Rational and Irrational Numbers. . . . . . . . . . 1

1.3 Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 The Completeness Axiom. . . . . . . . . . . . . . . . . . . 5

1.6 An Axiomatic System for the Set of Real Numbers. . . . . 5

1.7 The Absolute Value. . . . . . . . . . . . . . . . . . . . . . 7

2 Sequences and Accumulation Points 9

2.1 Convergent Sequences. . . . . . . . . . . . . . . . . . . . 9

2.2 Arithmetic of Converging Sequences. . . . . . . . . . . . . 12

2.3 Monotone Sequences. . . . . . . . . . . . . . . . . . . . . 13

2.4 Subsequences. . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Limes Inferior and Limes Superioropt . . . . . . . . . . . . . 18

2.6 Cauchy Sequences. . . . . . . . . . . . . . . . . . . . . . 19

2.7 Accumulation Points. . . . . . . . . . . . . . . . . . . . . 20

3 Limits 23

3.1 Definition and Examples. . . . . . . . . . . . . . . . . . . 23

3.2 Arithmetic of Limitsopt . . . . . . . . . . . . . . . . . . . . 27

3.3 Monotone Functions. . . . . . . . . . . . . . . . . . . . . 28

4 Set Theoryopt 31

4.1 Cardinality . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 The Cantor Set. . . . . . . . . . . . . . . . . . . . . . . . 34

5 Continuity 37


5.2 Combinations of Continuous Functions. . . . . . . . . . . 39

5.3 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . 40

5.4 Continuous Functions on Closed Intervals. . . . . . . . . . 42

6 The Derivative 45


6.2 Techniques of Differentiation. . . . . . . . . . . . . . . . . 46

6.3 The Mean-Value Theorem and its Applications. . . . . . . 47

6.4 The Derivative and the Intermediate Value Propertyopt . . . . 50

7 The Integral [Under Construction] 53


7.2 Arithmetic of Integrals . . . . . . . . . . . . . . . . . . . . 56

7.3 The Fundamental Theorem of Calculus. . . . . . . . . . . 57

A Hints and Remarks 59

B Greek Letters 65

C References for Additional Reading 67

List of Figures

1 (i) A sequence(xn) converges to the limita if . . . (ii) . . . forall ε > 0 . . . (iii) . . . there is anN ∈ N, such that . . . (iv). . . |xn − a| < ε for all n ≥ N . . . . . . . . . . . . . . . . . 11

2 A divergent sequence.. . . . . . . . . . . . . . . . . . . . . 12

3 The graph of the function in Exercise56 . . . . . . . . . . . 23

4 The graph ofx sin (1/x) . . . . . . . . . . . . . . . . . . . 24

5 The graph ofsin (1/x) . . . . . . . . . . . . . . . . . . . . 25

6 The graph of the function in Task60. . . . . . . . . . . . . . 26

7 The first steps in building the Cantor set.. . . . . . . . . . . 35

8 The Intermediate Value Theorem. . . . . . . . . . . . . . . 43

9 The graph ofx2 sin (1/x) . . . . . . . . . . . . . . . . . . . 46

10 The Mean Value Theorem. . . . . . . . . . . . . . . . . . 48

11 A partition of the interval[a, b]. . . . . . . . . . . . . . . . . 53

12 A lower Riemann sum with a partition consisting of 20 equally-spaced points.. . . . . . . . . . . . . . . . . . . . . . . . . 54

13 An upper Riemann sum with a partition consisting of 12 equally-spaced points.. . . . . . . . . . . . . . . . . . . . . . . . . 54

1 Introduction

When we want to study a subject in Mathematics, we first have to agreeupon what we assume we all already understand.

In this course we will assume that we are all familiar with the Real Numbers.Before we list the basic axioms the Real Numbers satisfy, we will brieflyreview more elementary concepts of numbers.

1.1 The Set of Natural Numbers

When we start learning Mathematics in elementary school, we live in theworld of NATURAL NUMBERS, which we will denote byN:

N = 1, 2, 3, 4, . . .

Natural numbers are the “natural” objects to count things around us with.The first thing we learn is to add natural numbers, then later on we start tomultiply.

We will take the following characterization of the Natural NumbersN forgranted throughout the course:

Axiom N1 1 ∈ N.

Axiom N2 If n ∈ N, thenn+ 1 ∈ N.

Axiom N3 There is no natural numbern ∈ N, such thatn+ 1 = 1.

We will also assume the following axiom, even though it can be deducedfrom the Completeness Axiom of the Real Numbers (see Optional Task28):

Axiom N4 For every natural numbern ∈ N, there is a real numberr ∈ Rsuch thatn ≤ r < n+ 1.

1.2 Integers, Rational and Irrational Numbers

Deficiencies of the system of natural numbers start to appear when we wantto divide—the quotient of two natural numbers is not necessarily a natural

2 Introduction

number, or when we want to subtract—the difference of two natural num-bers is not necessarily a natural number. This leads quite naturally to twoextensions of the concept of number.

The set ofINTEGERS, denoted byZ, is the set

Z = 0, 1,−1, 2,−2, 3,−3, . . ..

The set ofRATIONAL NUMBERS Q is defined as

Q =

p

q| p, q ∈ Z andq 6= 0

.

Real numbers that are not rational are calledIRRATIONAL NUMBERS. Theexistence of irrational numbers, first discovered by the Pythagoreans inabout 520 B.C., must have come as a major surprise to Greek Mathemati-cians:

Task 1 The square root of2 is irrational. (√

2 is the positive numberwhose square is2.)

1.3 Groups

Next we will put the properties of addition and multiplication of real num-bers into a wider context by introducing the concept of a “abelian group”and, in the next section, the concept of a “field”.

A setG with a binary operation∗ is called anABELIAN GROUP, if (G, ∗)satisfies the following axioms:

G1 ∗ is a well-defined map fromG×G toG.

G2 (Associativity) For alla, b, c ∈ G

(a ∗ b) ∗ c = a ∗ (b ∗ c)

G3 (Existence of a neutral element)There is an elementn ∈ G such thatfor all a ∈ G

a ∗ n = a

1.4 Fields 3

G4 (Existence of inverse elements)For everya ∈ G there existsb ∈ Gsuch that

a ∗ b = n

G5 (Commutativity) For alla, b ∈ G

a ∗ b = b ∗ a

The setsZ,Q andR are examples of abelian groups when endowed with theusual addition+ . The natural element in these cases is0; it is customary todenote the inverse element ofa by−a.

Exercise 2 Write down the axiomsG1–G5 explicitly for the group(Z,+).

The setsQ \ 0 and R \ 0 also form abelian groups under the usualmultiplication · . In these cases we denote the natural element by1; theinverse element ofa is customarily denoted by1/a or bya−1.

Exercise 3 Write down the axiomsG1–G5 explicitly for the group(Q \ 0, ·).

Addition and multiplication of rational and real numbers interact in a rea-sonable manner—the followingDISTRIBUTIVE LAW holds:

DL For alla, b, c ∈ R

(a+ b) · c = (a · c) + (b · c)

1.4 Fields

In short, a setF together with an addition+ and a multiplication· is calleda FIELD, if

F1 (F,+) is an abelian group (with neutral element0).

4 Introduction

F2 (F \ 0, ·) is an abelian group (with neutral element1).

F3 For alla, b, c ∈ F : (a+ b) · c = (a · c) + (b · c).

The rational numbers and the real numbers are examples of fields.

Another example of a field is the set of complex numbersC.

C = a+ bi | a, b ∈ R

Addition and multiplication of complex numbers are defined as follows:

(a+ bi) + (c+ di) = (a+ c) + (b+ d)i,

and

(a+ bi) · (c+ di) = (ac− bd) + (ad+ bc)i,

respectively.

A field F endowed with a relation< is called anORDERED FIELDif

O1 For allx, y, z ∈ F

x < y impliesx+ z < y + z

O2 For allx, y ∈ F and allz > 0

x < y impliesx · z < y · z

O3 For allx, y, z ∈ F

x < y andy < z impliesx < z

O4 For allx, y ∈ Fx < y, y < x, or x = y

Both the rational numbersQ and the real numbersR form ordered fields.The complex numbersC cannot be ordered in such a way.

1.5 The Completeness Axiom 5

1.5 The Completeness Axiom

You probably have seen books entitled “Real Analysis” and “Complex Anal-ysis” in the library. There are no books on “Rational Analysis”.

Why? What is the main difference between the two ordered fields ofQ andR?—The ordered fieldR of real numbers isCOMPLETE: sequences of realnumbers have the following property.

C Let (an) be an increasing sequence of real numbers. If(an) is boundedfrom above, then(an) converges.

The ordered fieldQ of rational numbers, on the other hand, isnot complete.It should therefore not surprise you that the Completeness Axiom will playa central part throughout the course! We will discuss this axiom in greatdetail in Section2.3.

The complex numbersC also form a complete field. Section2.3will give ahint on how to write down an appropriate completeness axiom for the fieldC.

1.6 An Axiomatic System for the Set of Real Numbers

Below is a summary of the properties of the real numbersR we will take forgranted throughout the course:

The set of real numbersR with its natural operations of+, ·, and< forms acomplete ordered field. This means that the real numbers satisfy the follow-ing axioms:

Axiom 1 + is a well-defined map fromR× R to R.

Axiom 2 For alla, b, c ∈ R

(a+ b) + c = a+ (b+ c)

Axiom 3 There is an element0 ∈ R such that for alla ∈ R

a+ 0 = a

Axiom 4 For everya ∈ R there existsb ∈ R such that

a+ b = 0

6 Introduction

Axiom 5 For alla, b ∈ Ra+ b = b+ a

Axiom 6 · is a well-defined map fromR \ 0 × R \ 0 to R \ 0.

Axiom 7 For alla, b, c ∈ R \ 0

(a · b) · c = a · (b · c)

Axiom 8 There is an element1 ∈ R \ 0 such that for alla ∈ R \ 0

a · 1 = a

Axiom 9 For everya ∈ R \ 0 there existsb ∈ R \ 0 such that

a · b = 1

Axiom 10 For alla, b ∈ R \ 0

a · b = b · a

Axiom 11 For alla, b, c ∈ R

(a+ b) · c = (a · c) + (b · c)

Axiom 12 For allx, y, z ∈ R

x < y impliesx+ z < y + z

Axiom 13 For allx, y ∈ R and allz > 0

x < y impliesx · z < y · z

Axiom 14 For allx, y, z ∈ R

x < y andy < z impliesx < z

Axiom 15 For allx, y ∈ R

x < y, y < x, or x = y

Axiom 16 Let (an) be an increasing sequence of real numbers. If(an) isbounded from above, then(an) converges.

1.7 The Absolute Value 7

1.7 The Absolute Value

TheABSOLUTE VALUE of a real numbera is defined as

|a| = maxa,−a.

For instance,|4| = 4, |−π| = π. The quantity|a− b|measures the distanceon the real number line between two real numbersa andb.

The following result is known as thetriangle inequality :

Exercise 4 For alla, b ∈ R:

|a+ b| ≤ |a|+ |b|

A related result is called thereverse triangle inequality:

Exercise 5 For alla, b ∈ R:

|a− b| ≥∣∣∣∣ |a| − |b|

∣∣∣∣

8 Introduction

2 Sequences and Accumulation Points

2.1 Convergent Sequences

Formally, aSEQUENCE OF REAL NUMBERSis a functionϕ : N → R. For

instance the functionϕ(n) =1

n2for all n ∈ N defines a sequence. It is

customary, though, to write sequences by listing their terms such as

1,1

4,1

9,

1

16, . . .

or by writing them in the form(an)n∈N; so in our particular example we

could write the sequence also as

(1

n2

)n∈N

. Note that the elements of a

sequence come in a natural order. For instance1

49is the 7th element of the

sequence

(1

n2

)n∈N

.

Exercise 6 Let (an)n∈N denote the sequence of prime numbers in theirnatural order. What isa5? (“a5” is pronounced “a sub 5”.)

Exercise 7 Write the sequence0, 1, 0, 2, 0, 3, 0, 4, . . . as a functionϕ :N → R.

We say that a sequence(an) IS CONVERGENT, if there is a real numbera,such that for allε > 0 there is anN ∈ N such that for alln ∈ N withn ≥ N ,

|an − a| < ε.

The numbera is calledLIMIT of the sequence(an). We also say in this casethat the sequence(an)n∈N CONVERGES TOa.


Exercise 8 Spend some quality time studying Figure1 on page11.Explain how the pictures and the parts in the definition correspond to eachother. Also reflect on how the “rigorous” definition above relates to yourprior understanding of what it means for a sequence to converge.

A sequence, which fails to converge, is calledDIVERGENT. Figure2 onpage12gives an example.

Exercise 9 Write down (using theε-N language) what it means thata given sequence(an)n∈N does not converge to the real numbera. Writedown formally what it means that a sequence diverges.

Exercise 10 Show that the sequencean =(−1)n

√n

converges to0.

Exercise 11 Show that the sequencean = 1− 1

n2 + 1converges to1.

The first result establishes that limits are unique.

Task 12 If a sequence converges to two real numbersa and b, thena = b.

We say that a setS of real numbers isBOUNDED if there are real numbersm andM such that

m ≤ s ≤M

holds for alls ∈ S.

2.1 Convergent Sequences 11

Figure 1: (i) A sequence(xn) converges to the limita if . . . (ii) . . . for allε > 0 . . . (iii) . . . there is anN ∈ N, such that . . . (iv) . . .|xn − a| < ε forall n ≥ N .


Figure 2: A divergent sequence.

A sequence(an) is calledBOUNDED if its range

an | n ∈ N

is a bounded set.

Exercise 13 Give an example of an unbounded sequence.

Exercise 14 Give an example of a bounded sequence which does notconverge.

Task 15 Every convergent sequence is bounded.

Consequently, boundedness is necessary for convergence of a sequence, butis not sufficient to ensure that a sequence is convergent.

2.2 Arithmetic of Converging Sequences

The following results deal with the “arithmetic” of convergent sequences.

2.3 Monotone Sequences 13

Task 16 If the sequence(an) converges toa, and the sequence(bn)converges tob, then the sequence(an + bn) is also convergent and its limitis a+ b.

Task 17 If the sequence(an) converges toa, and the sequence(bn)converges tob, then the sequence(an · bn) is also convergent and its limit isa · b.

Task 18 Let (an) be a sequence converging toa 6= 0. Then there are aδ > 0 and anM ∈ N such that|am| > δ for all m ≥M .

Task18 is useful in proving:

Task 19 Let the sequence(bn) with bn 6= 0 for all n ∈ N converge to

b 6= 0. Then the sequence

(1

bn

)is also convergent and its limit is

1

b.

Task 20 Let (an) be a sequence converging toa. If an ≥ 0 for alln ∈ N, thena ≥ 0.

2.3 Monotone Sequences

Let A be a non-empty set of real numbers. We say thatA is BOUNDED

FROM ABOVE if there is anM ∈ R such thata ≤ M for all a ∈ A. M isthen called anUPPER BOUNDfor A. Similarly, we say thatA is BOUNDED


FROM BELOW if there is anm ∈ R such thata ≥ m for all a ∈ A. m isthen called aLOWER BOUND for A.

A sequence is bounded from above (bounded from below), if its range isbounded from above (bounded from below).

A sequence(an) is calledINCREASING if am ≤ an for all m < n ∈ N. It iscalledSTRICTLY INCREASING if am < an for all m < n ∈ N. A sequence(an) is calledDECREASING if am ≥ an for all m < n ∈ N. It is calledSTRICTLY DECREASINGif am > an for all m < n ∈ N. A sequence whichis increasing or decreasing is calledMONOTONE.

The following axiom is a fundamental property of the real numbers. It es-tablishes that bounded monotone sequences are convergent. This result isof fundamental importance for the study of series.

Completeness Axiom of the Real Numbers.Let (an) be anincreasing bounded sequence. Then(an) converges.

The same result holds of course if one replaces “increasing” and “boundedfrom above” by “decreasing” and “bounded from below”.

Note that an increasing sequence is always bounded from below, while adecreasing sequence is always bounded from above.

Task 21 Let a1 = 1 andan+1 =√

3an for all n ∈ N. Show that thesequence(an) converges.

Let A be a non-empty set of real numbers. We say that a real numbers isthe LEAST UPPER BOUNDof A (or thats is theSUPREMUM of A), if s isan upper bound ofA and if no number smaller thans is an upper boundfor A. We writes = supA. Similarly, we say that a real numberi is theGREATEST LOWER BOUNDof A (or thati is the INFIMUM of A), if i is alower bound ofA and if no number greater thani is a lower bound forA.We writei = inf A.

Exercise 22 Find the supremum of each of the following sets:

1. The closed interval[−2, 3]

2.3 Monotone Sequences 15

2. The open interval (0,2)

3. The setx ∈ Z | x2 < 5

4. The setx ∈ Q | x2 < 3.

Exercise 23 Show that an increasing bounded sequence converges tothe supremum of its range.

Task 24 Assume the non-empty setS is bounded from above. Showthat for allε > 0 there is an elementa ∈ S such thata+ε is an upper boundfor S.

The result in Task24 is helpful in proving the “hard” direction of the taskbelow.

Task 25 The Completeness Axiom is equivalent to the following: Ev-ery non-empty set of real numbers which is bounded from above has a supre-mum.

We say that a setA of real numbers isDENSE in R, if for all real numbersx < y there is an elementa ∈ A satisfyingx < a < y.

Task 26 The set of rational numbersQ is dense inR.

Task 27 The set of irrational numbersR \Q is dense inR.


Optional Task 28 Use the Completeness Axiom to show the follow-ing: For every natural numbern ∈ N, there is a real numberr ∈ R such thatn ≤ r < n+ 1.

2.4 Subsequences

Recall that a sequence is a mapϕ : N → R. Let ψ : N → N be a strictlyincreasing map. (A mapψ : N → N is called strictly increasing if it satisfies:ψ(n) < ψ(m) for all n < m in N).

Then the sequenceϕ ψ : N → R is called aSUBSEQUENCEof ϕ.

Here is an example: Suppose we are given the sequence

1,1

2,1

3,1

4,1

5,1

6,1

7,1

8, . . .

The mapψ(n) = 2n then defines the subsequence

1

2,1

4,1

6,1

8, . . .

If we denote the original sequence by(an), and ifψ(k) = nk for all k ∈ N,then we denote the subsequence by(ank

).

So, in the example above,

an1 = a2 =1

2, an2 = a4 =

1

4, an3 = a6 =

1

6, . . .

Exercise 29 Let (an)n∈N =

(1

n

)n∈N

. Which of the following se-

quences are subsequences of(an)n∈N?

1. 1,1

2,1

3,1

4,1

5. . .

2.1

2, 1,

1

4,1

3,1

6,1

5. . .

2.4 Subsequences 17

3. 1,1

3,1

6,

1

10,

1

15. . .

4. 1, 1,1

3,1

3,1

5,1

5. . .

For the subsequence examples, also find the functionψ : N → N.

Task 30 Let (an)n∈N be a sequence of real numbers, and let(ank)k∈N

be one of its subsequences. What does it meanformally that (ank) con-

verges toa ∈ R?Show that the formal definition is equivalent to the following: For allε > 0there exists anN ∈ N such that|ank

− a| < ε for all nk ≥ N .

Task 31 If a sequence converges, then all of its subsequences convergeto the same limit.

Task 32 Every sequence of real numbers has an increasing subse-quence or it has a decreasing subsequence.

The next fundamental result is known as theBolzano-Weierstrass Theo-rem.

Task 33 Every bounded sequence of real numbers has a convergingsubsequence.


Task 34 Suppose the sequence(an) does not converge to the real num-berL. Then there is anε > 0 and a subsequence(ank

) of (an) such that

|ank− L| ≥ ε for all k ∈ N.

We conclude this section with a rather strange result: it establishes con-vergence of a bounded sequence without ever showing any convergence atall.

Task 35 A boundedsequence converges if all of itsconvergentsub-sequences converge to the same limit.

2.5 Limes Inferior and Limes Superioropt

Let (an) be a bounded sequence of real numbers. We define theLIMES

INFERIOR andLIMES SUPERIORof the sequence as

lim infn→∞

an := limk→∞

(infan | n ≥ k) ,

andlim sup

n→∞an := lim

k→∞(supan | n ≥ k) .

Optional Task 36 Explain why the numbers lim infn→∞

an and

lim supn→∞

an are well-defined for every bounded sequence(an).

Optional Task 37 Show that a bounded sequence(an) converges ifand only if

lim infn→∞

an = lim supn→∞

an.

2.6 Cauchy Sequences 19

Optional Task 38 Let (an) be a bounded sequence of real numbers.Show that(an) has a subsequence that converges tolim sup

n→∞an.

Optional Task 39 Let (an) be a bounded sequence of real numbers,and let(ank

) be one of its converging subsequences. Show that

lim infn→∞

an ≤ limk→∞

ank≤ lim sup

n→∞an.

Optional Task 40 Let an := sinn for all n ∈ N. Show thatlim sup

n→∞an = 1. (You may use prior knowledge from your Calculus classes.)

2.6 Cauchy Sequences

A sequence(an)n∈N is called a CAUCHY SEQUENCE, if for all ε > 0 thereis anN ∈ N such that for allm,n ∈ N with m ≥ N andn ≥ N ,

|am − an| < ε.

Informally speaking: a sequence is convergent, if far out all terms of thesequence are close to the limit; a sequence is a Cauchy sequence, if far outall terms of the sequence are close to each other.

We will establish in this section that a sequence converges if and only if itis a Cauchy sequence.

You may wonder why we bother to explore the concept of a Cauchy se-quence when it turns out that Cauchy sequences are nothing else but conver-gent sequences. Answer: You can’t show directly that a sequence convergeswithout knowing its limit a priori. The concept of a Cauchy sequence onthe other hand allows you to show convergence without knowing the limitof the sequence in question! This will nearly always be the situation when


you study series of real numbers. The “Cauchy criterion” for series turnsout to be one of most widely used tools to establish convergence of series.

Exercise 41 Every convergent sequence is a Cauchy sequence.

Exercise 42 Every Cauchy sequence is bounded.

Task 43 If a Cauchy sequence has a converging subsequence with limita, then the Cauchy sequence itself converges toa.

Task 44 Every Cauchy sequence is convergent.

2.7 Accumulation Points

Givenx ∈ R andε > 0, we say that the open interval(x− ε, x + ε) formsa NEIGHBORHOOD OFx.

Task 45 A sequence(an) converges toL ∈ R if and only if everyneighborhood ofL contains all but a finite number of the terms of the se-quence(an).

The real numberx is called anACCUMULATION POINT of the setS, if everyneighborhood of x contains infinitely many elements ofS.

2.7 Accumulation Points 21

Task 46 The real numberx is an accumulation point of the setS if andonly if every neighborhood ofx contains an element ofS different fromx.

Note that finite sets do not have accumulation points. The following exerciseprovides some more examples:

Exercise 47 Find all accumulation points of the following sets:

1. Q

2. N

3. [a, b)

4.

1

n| n ∈ N

.

Exercise 48 Find a set of real numbers with exactly two accumulationpoints.

Exercise 49 Find a set of real numbers whose accumulation pointsform a sequence(an) with an 6= am for all n 6= m.

Task 50 Show thatx is an accumulation point of the setS if and onlyif there is a sequence(xn) of elements inS with xn 6= xm for all n 6= msuch that(xn) converges tox.


Task 51 Every infinite bounded set of real numbers has at least oneaccumulation point.

Task 52 Let S be a non-empty set of real numbers which is boundedfrom above. Show: IfsupS 6∈ S, thensupS is an accumulation point ofS.

The remaining tasks in this section explore the relationship between thelimit of a converging sequence and accumulation points of its range.

Optional Task 53 Find a converging sequence whose range has ex-actly one accumulation point. Find a converging sequence whose range hasno accumulation points.

Optional Task 54 The range of a converging sequence has at mostone accumulation point.

Optional Task 55 Suppose the sequence(an) is bounded and satis-fies the condition thatam 6= an for allm 6= n ∈ N. If its rangean | n ∈ Nhas exactly one accumulation pointa, then(an) converges toa.

3 Limits

3.1 Definition and Examples

Let D ⊆ R, let f : D → R be a function and letx0 be an accumulationpoint ofD.

We say that theLIMIT of f(x) atx0 is equal toL ∈ R, if for all ε > 0 thereis aδ > 0 such that

|f(x)− L| < ε

wheneverx ∈ D and0 < |x− x0| < δ.

In this case we writelimx→x0

f(x) = L.

Note that—by design—the existence of the limit (andL itself) does notdepend on what happens whenx = x0, but only on what happens “close” tox0.

Figure 3: The graph of the function in Exercise56

Exercise 56 Let f : R → R be defined by

f(x) =

|x|/x, if x 6= 0, x ∈ R

0, if x = 0

24 Limits

Doesf(x) have a limit ax0 = 0? If so, what is the limit?See Figure3 on page23.

In the next two problems you may assume the usual properties of the sinefunction.


f(x) =

x sin

(1x

), if x 6= 0, x ∈ R

0, if x = 0


Figure 4: The graph ofx sin (1/x)


f(x) =

sin(

1x

), if x 6= 0, x ∈ R

0, if x = 0

3.1 Definition and Examples 25


Figure 5: The graph ofsin (1/x)

Exercise 59 Let f : (0, 1] → R be defined by

f(x) =

1, if x ∈ Q0, if x ∈ R \Q

For which values ofx0 doesf(x) have a limit ax0? What is the limit?

Task 60 Let f : (0, 1] → R be defined by

f(x) =

1

q, if x =

p

qwith p, q relatively prime positive integers

0, if x ∈ R \Q

For which values ofx0 doesf(x) have a limit ax0? What is the limit?See Figure6 on page26.

26 Limits

Figure 6: The graph of the function in Task60.

The next two problems reduce the study of the concept of a limit of a func-tion at a point to our earlier study of sequence convergence.

Exercise 61 LetD ⊆ R, let f : D → R be a function and letx0 be anaccumulation point ofD. Then the following are equivalent:

1. limx→x0

f(x) exists and is equal toL.

2. Let (xn) be a sequence of elements inD that converges tox0, and sat-isfies thatxn 6= x0 for all n ∈ N. Then the sequencef(xn) convergestoL.

Optional Task 62 LetD ⊆ R, let f : D → R be a function and letx0 be an accumulation point ofD. Then the following are equivalent:

1. limx→x0

f(x) exists.

2. Let (xn) be a sequence of elements inD that converges tox0, and sat-isfies thatxn 6= x0 for all n ∈ N. Then the sequencef(xn) converges.

3.2 Arithmetic of Limits opt 27

The result below is called thePrinciple of Local Boundedness.

Exercise 63 LetD ⊆ R, let f : D → R be a function and letx0 be anaccumulation point ofD.If f(x) has a limit atx0, then there is aδ > 0 and anM > 0 such that

|f(x)| ≤M for all x ∈ (x0 − δ, x0 + δ) ∩D.

3.2 Arithmetic of Limits opt

Optional Task 64 LetD ⊆ R, let f, g : D → R be functions and letx0 be an accumulation point ofD.If lim

x→x0

f(x) = L and limx→x0

g(x) = M , then the sumf + g has a limit atx0,

and limx→x0

(f + g)(x) = L+M .

Optional Task 65 LetD ⊆ R, let f, g : D → R be functions and letx0 be an accumulation point ofD.If lim

x→x0

f(x) = L and limx→x0

g(x) = M , then the productf · g has a limit at

x0, and limx→x0

(f · g)(x) = L ·M .

Optional Task 66 LetD ⊆ R, let f : D → R be a function and letx0 be an accumulation point ofD. Assume additionally thatf(x) 6= 0 forall x ∈ D.If lim

x→x0

f(x) = L and ifL 6= 0, then the reciprocal function1/f has a limit

atx0, and limx→x0

1

f(x)=

1

L.

28 Limits

3.3 Monotone Functions

Let a < b be real numbers. A functionf : [a, b] → R is calledINCREASING

on [a, b], if x < y implies f(x) ≤ f(y) for all x, y ∈ [a, b]. It is calledSTRICTLY INCREASING on [a, b], if x < y implies f(x) < f(y) for allx, y ∈ [a, b].

Similarly, a functionf : [a, b] → R is calledDECREASING on [a, b], ifx < y implies f(x) ≥ f(y) for all x, y ∈ [a, b]. It is called STRICTLY

DECREASINGon [a, b], if x < y impliesf(x) > f(y) for all x, y ∈ [a, b].

A function f : [a, b] → R is calledMONOTONE on [a, b] if it is increasingon [a, b] or it is decreasing on[a, b].

As we have seen in the last section, a function can fail to have limits forvarious reasons. Monotone functions, on the other hand, are easier to un-derstand: a monotone function fails to have a limit at a point if and only ifit “jumps” at that point. The next task makes this precise.

Task 67 Let a < b be real numbers and letf : [a, b] be an increasingfunction. Letx0 ∈ (a, b). We define

L(x0) = supf(y) | y ∈ [a, x0)

andU(x0) = inff(y) | y ∈ (x0, b]

Thenf(x) has a limit atx0 if and only ifU(x0) = L(x0). In this case

U(x0) = L(x0) = f(x0) = limx→x0

f(x).

Task 68 Under the assumptions of the previous task, state and prove aresult discussing the existence of a limit at the endpointsa andb.

Task 69 Let a < b be real numbers and letf : [a, b] be an increasingfunction. Show that the set

y ∈ [a, b] | f(x) does not have a limit aty

3.3 Monotone Functions 29

is finite or countable. (This means that the set can be arranged as a finite orinfinite sequencey1, y2, y3, . . .).

30 Limits

4 Set Theoryopt

4.1 Cardinality

We say that two setsA andB have the sameCARDINALITY if there is abijection (a 1-1 and onto map)ϕ : A → B. If A andB have the samecardinality we writeA ∼ B.

Given a setG its POWER SETP(G) is the set of all subsets ofG:

P(G) := A | A ⊆ G

For instance,P(1, 2) = ∅, 1, 2, 1, 2.

Optional Task 70 LetG be some set. Then “∼” defines an equiva-lence relation on the power set ofG, i.e., the following holds:

1. A ∼ A for all A ∈ P(G)

2. If A ∼ B, thenB ∼ A for all A,B ∈ P(G)

3. If A ∼ B andB ∼ C, thenA ∼ C holds for allA,B,C ∈ P(G)

Let n ∈ N andA be a set. IfA ∼ 1, 2, 3, . . . , n, we sayA has cardinal-ity n. Sets with cardinalityn for somen ∈ N are calledFINITE. The emptyset∅ is said to have cardinality0 and is also considered to be a finite set.

A setA, for whichA ∼ N is calledCOUNTABLE, or said to haveCOUNT-ABLE CARDINALITY .

The next result is attributed to Galileo Galilei (1564–1642):

Optional Task 71 Let 2N be the set of even natural numbers. Show:N ∼ 2N.

Optional Task 72 N ∼ Z.

32 Set Theoryopt

The set of rational numbers is also countable; this result like all the follow-ing results in this section were discovered by Georg Cantor (1845–1918).

Optional Task 73 N ∼ Q.

Are there uncountable sets? The answer to this question is a loud yes! Thenext result establishes thatP(N) is an example of an infinite set which isnot countable.

This result, known asCantor’s Theorem, is considered one of the mostbeautiful results in all of mathematics. It generalizes the well-known factthat for finite setsG of cardinalityn, the power setP(G) has cardinality2n.

Optional Task 74 Let A be any set. ThenP(A) does not have thesame cardinality asA.

Cantor’s Theorem shows that there is at least a countable number of distinctuncountable cardinalities.

The rest of this section is devoted to a proof of the fact thatR ∼ P(N), thusestablishing that the set of real numbers is uncountable.

Optional Task 75 R ∼ (0, 1).

Optional Task 76 (0, 1) ∼ [0, 1].

It is left to establish that[0, 1] ∼ P(N). To see this we will write elementsx ∈ [0, 1] in their BINARY EXPANSION: Everyx ∈ [0, 1] can be written as

x =∞∑

n=1

εn

2n

4.1 Cardinality 33

with εn either 0 or 1 for alln ∈ N.

For instance5

8=

1

21+

0

22+

1

23and1 =

1

2+

1

22+

1

23+

1

24+ . . .

Optional Task 77 1

3=

1

22+

1

24+

1

26+

1

28+ . . .

The next two tasks address a minor technical problem: the binary expansionof a real number is not always unique, some numbers in[0, 1] have twodifferent binary expansions.

Optional Task 78 Let B be the set of those real numbers in[0, 1]with two distinct binary expansions.

1. Find an element inB.

2. Classify all real numbers inB.

3. Show that the setB is countable (and infinite).

Optional Task 79 SupposeA ∼ P(N), and letB ⊆ A with B ∼ N.Then

(A \B) ∼ P(N).

Optional Task 80 [0, 1] ∼ P(N).

Is it true that every setA of real numbers containingN either has the cardi-nality of N or the cardinality ofR? This question is known as the CONTIN-UUM HYPOTHESIS:

DoesN ⊂ A ⊂ R imply that eitherA ∼ N orA ∼ R?

34 Set Theoryopt

The Continuum Hypothesis cannot be answered. Within the usual axiomaticsystem of set theory, no contradictions arise from either a positive or a neg-ative answer to the Continuum Hypothesis. This deep result that the Con-tinuum Hypothesis is independent of the axioms of set theory was provedby Paul Cohen (1934–) in 1963.

4.2 The Cantor Set

A non-empty set is calledPERFECT if it equals its set of accumulationpoints.R itself and closed intervals are examples of perfect sets.

Optional Task 81 Every perfect set is uncountable.

Note that this result provides a second proof of the fact thatR is uncount-able, without giving the additional information thatR ∼ P(N).

We now construct a more interesting perfect set: LetC0 = [0, 1]. Let’sremove the middle third ofC0:

C1 =

[0,

1

3

]∪[2

3, 1

].

Next we remove the two middle thirds ofC1:

C2 =

[0,

1

9

]∪[2

9,1

3

]∪[2

3,7

9

]∪[8

9, 1

].

Continue in this fashion: in each step remove the middle thirds of the inter-vals from the previous step.

See Figure7.

The CANTOR SETC is defined as the intersection of all these sets:

C =∞⋂

n=0

Cn.

4.2 The Cantor Set 35

Figure 7: The first steps in building the Cantor set.

Optional Task 82 Show thatC is perfect.

Optional Task 83 Show thatC does not contain an open interval.

What is the “length” of the Cantor set? ClearlyC0 has length 1 (being aninterval of length 1).

Similarly, C1 has length23, C2 has length

22

32, etc. SinceC is contained in

all Cn, it must have “length” 0.

The notion of “length” will be made precise in a course on measure theory.You will learn in such a course thatC is indeed a set of measure 0.

36 Set Theoryopt

Optional Task 84 The Cantor set consists of all real numbers in[0, 1], which have a base-3 expansion with digits0 and2:

C =

∞∑

n=1

εn

3n| εn ∈ 0, 2 for all n ∈ N

.

Task 84 provides an alternative way to show thatC is uncountable, andindeed thatC has the same cardinality asR. Letϕ : [0, 1] → C, be definedby

ϕ

(∞∑

n=1

εn

2n

)=

∞∑n=1

2εn

3n.

Here, every real numberx ∈ [0, 1] is written in its binary expansion as

x =∞∑

n=1

εn

2n

with εn ∈ 0, 1 for all n ∈ N. The map “essentially” is a bijection; there isa small problem again with those real numbers who have two distinct binaryexpansions.

5 Continuity


LetD be a set of real numbers andx0 ∈ D. A functionf : D → R is saidto be CONTINUOUS at x0 if the following holds: For allε > 0 there is aδ > 0 such that for allx ∈ D with

|x− x0| < δ,

we have that|f(x)− f(x0)| < ε.

If the function is continuous at allx0 ∈ D, we simply say thatf : D → Ris continuous onD.

Compare this definition of continuity to the earlier definition of having alimit. For continuity, we want to ensure that the behavior of the functionclose to the pointx0 nicely interacts with the behavior of the function at thepoint in question itself; thus we require thatx0 lies in the domainD. Notealso that we do no longer require in the definition of continuity thatx0 is anaccumulation point ofD.

Exercise 85 Let D be a set of real numbers andx0 ∈ D be an accu-mulation point ofD. The functionf : D → R is continuous atx0 if andonly if lim

x→x0

f(x) = f(x0).

Exercise 86 LetD be a set of real numbers andx0 ∈ D. Assume alsothatx0 is not an accumulation point ofD. Then the functionf : D → R iscontinuous atx0.

Exercise 87 LetD be a set of real numbers andx0 ∈ D. A functionf : D → R is continuous atx0 if and only if for all sequences(xn) in Dconverging tox0, the sequence(f(xn)) converges tof(x0).

38 Continuity


f(x) =

|x|, if x ∈ Qx2, if x ∈ R \Q

For which values ofx0 is f(x) continuous?


f(x) =

x sin

(1x

), if x 6= 0, x ∈ R

0, if x = 0

Is f(x) continuous atx0 = 0?See Figure4 on page24.


f(x) =

sin(

1x

), if x 6= 0, x ∈ R

0, if x = 0

Is f(x) continuous atx0 = 0?See Figure5 on page25.


f(x) =

1, if x ∈ Q0, if x ∈ R \Q

For which values ofx0 is f(x) continuous?

5.2 Combinations of Continuous Functions 39


f(x) =

1

q, if x =

p

qwith p, q relatively prime positive integers

0, if x ∈ R \Q

For which values ofx0 is f(x) continuous?See Figure6 on page26.

5.2 Combinations of Continuous Functions

Optional Task 93 Let D ⊆ R, let f, g : D → R be functions con-tinuous atx0 ∈ D. Thenf + g : D → R is continuous atx0.

Optional Task 94 Let D ⊆ R, let f, g : D → R be functions con-tinuous atx0 ∈ D. Thenf · g : D → R is continuous atx0.

Optional Task 95 Polynomials are continuous onR.

Optional Task 96 LetD ⊆ R, let f : D → R be a function contin-uous atx0 ∈ D. Assume additionally thatf(x) 6= 0 for all x ∈ D. Then1

f: D → R is continuous atx0.

40 Continuity

Task 97 Let D,E ⊆ R, let f : D → R be a function continuous atx0 ∈ D. Assumef(D) ⊆ E. Supposeg : E → R is a function continuousatf(x0). Then the compositiong f : D → R is continuous atx0.

5.3 Uniform Continuity

We say that a functionf : D → R is UNIFORMLY CONTINUOUSonD if thefollowing holds: For allε > 0 there is aδ > 0 such that wheneverx, y ∈ Dsatisfy

|x− y| < δ,

then|f(x)− f(y)| < ε.

Exercise 98 If f : D → R is uniformly continuous onD, thenf iscontinuous onD. What is the difference between continuity and uniformcontinuity?

Exercise 99 Let f : (0, 1) → R be defined byf(x) =1

x. Show that

f is not uniformly continuous on(0, 1).

Task 100 Let f : [a, b] → R be a continuous function on the closedinterval[a, b]. Thenf is uniformly continuous on[a, b].

In light of Exercise99, the result of Task100must depend heavily on prop-erties of closed bounded intervals. It is therefore natural to ask for whatdomains continuity automatically implies uniform continuity. The follow-ing two tasks explore this question.

5.3 Uniform Continuity 41

Optional Task 101 Let f : C → R be a continuous function on theCantor setC. Thenf is uniformly continuous onC.

Optional Task 102 Let D be a set of real numbers. Find sufficientconditions forD such that every continuous functionf : D → R is auto-matically uniformly continuous onD.

Task 103 Let f : D → R be uniformly continuous onD. If D is abounded subset ofR, thenf(D) is also bounded.

Optional Task 104 A functionf : (a, b) → R is uniformly continu-ous on the open interval(a, b) if and only if it can be defined at the endpointsa andb in such a way that the extensionf : [a, b] → R is continuous on theclosed interval[a, b].

Thus, for instance, the functionf : (0, 1) → R, given byf(x) = sin

(1

x

)is not uniformly continuous on(0, 1).

It is often easier to show uniform continuity by establishing the followingstronger condition:

A function f : D → R is called a LIPSCHITZ FUNCTIONonD if there isanM > 0 such that for allx, y ∈ D

|f(x)− f(y)| ≤M |x− y|

Exercise 105 Let f : D → R be a Lipschitz function onD. Thenfis uniformly continuous onD.

42 Continuity

Task 106 The functionf(x) =√x is uniformly continuous on the

interval[0, 1]. Show that it is not a Lipschitz function on the interval[0, 1].

5.4 Continuous Functions on Closed Intervals

The major goal of this section is to show that the continuous image of aclosed bounded interval is a closed bounded interval.

We say a functionf : D → R is BOUNDED, if there exists anM ∈ R suchthat|f(x)| ≤M for all x ∈ D.

Exercise 107 Let f : [a, b] → R be a continuous function on theclosed interval[a, b]. Thenf is bounded on[a, b].

We say that the functionf : D → R has anABSOLUTE MAXIMUM ifthere exists anx0 ∈ D such thatf(x) ≤ f(x0) for all x ∈ D. Similarly,f : D → R has anABSOLUTE MINIMUM if there exists anx0 ∈ D suchthatf(x) ≥ f(x0) for all x ∈ D.

We can improve upon the previous result as follows:

Task 108 Let f : [a, b] → R be a continuous function on the closedinterval[a, b]. Thenf has an absolute maximum (and an absolute minimum)on [a, b].

The next result is called theIntermediate Value Theorem.

Here the intervalI can be any interval. Also: Ifx > y, we understand theinterval(x, y) to be the interval(y, x).

5.4 Continuous Functions on Closed Intervals 43

Figure 8: The Intermediate Value Theorem

Task 109 Let f : I → R be a continuous function on the intervalI.Let a, b ∈ I. If d ∈ (f(a), f(b)), then there is a real numberc ∈ (a, b) suchthatf(c) = d.See Figure8 on page43.

A continuous function maps a closed bounded interval onto a closed boundedinterval:

Task 110 Let f : [a, b] → R be a continuous function on the closedinterval[a, b]. Thenf ([a, b]) := f(x) | x ∈ [a, b] is also a closed boundedinterval.

Task 111 Let f : [a, b] → R be strictly increasing (decreasing, resp.)and continuous on[a, b]. Thenf has an inverse onf([a, b]), which is strictlyincreasing (decreasing, resp.) and continuous.

44 Continuity

Task 112 Show that√x : [0,∞) → R is continuous on[0,∞).

6 The Derivative


LetD be a set of real numbers and letx0 ∈ D be an accumulation point ofD. The functionf : D → R is said to beDIFFERENTIABLE atx0, if

limx→x0

f(x)− f(x0)

x− x0

exists.

In this case, we call the limit above theDERIVATIVE of f atx0 and write

f ′(x0) = limx→x0

f(x)− f(x0)

x− x0

.

Exercise 113 Use the definition above to show that3√x : R → R is

differentiable on its domain.


f(x) =

x sin

(1x

), if x 6= 0, x ∈ R

0, if x = 0

Is f(x) differentiable atx0 = 0?See Figure4 on page24.


f(x) =

x2 sin

(1x

), if x 6= 0, x ∈ R

0, if x = 0

Is f(x) differentiable atx0 = 0? Using your Calculus knowledge, computethe derivative at pointsx0 6= 0. Is the derivative continuous atx0 = 0?See Figure9 on page46.

46 The Derivative

Figure 9: The graph ofx2 sin (1/x)

6.2 Techniques of Differentiation

Exercise 116 Supposef : D → R is differentiable atx0 ∈ D. Showthatf is continuous atx0.

Exercise 117 Give an example of a function with a point at whichfis continuous, but not differentiable.

Exercise 118 Let f, g : D → R be differentiable atx0 ∈ D. Then thefunctionf + g is differentiable atx0, with (f + g)′(x0) = f ′(x0) + g′(x0).

Next come some of the “Calculus Classics”, beginning with the “ProductRule”:

6.3 The Mean-Value Theorem and its Applications 47

Task 119 Let f, g : D → R be differentiable atx0 ∈ D. Then thefunctionf · g is differentiable atx0, with

(f · g)′(x0) = f ′(x0) · g(x0) + f(x0) · g′(x0).

In particular, ifc ∈ R, then

(c · f)′(x0) = c · f ′(x0).

Exercise 120 Show that polynomials are differentiable everywhere.Compute the derivative of a polynomial of the form

P (x) =n∑

k=0

akxk.

Task 121 State and prove the “Quotient Rule”.

Task 122 State and prove the “Chain Rule”.

6.3 The Mean-Value Theorem and its Applications

LetD be a subset ofR, and letf : D → R be a function. We say thatf hasa RELATIVE MAXIMUM atx0 ∈ D, if there is a neighborhoodU of x0, suchthat

f(x) ≤ f(x0) for all x ∈ U.Similarly, we say thatf has aRELATIVE MINIMUM atx0 ∈ D, if there is aneighborhoodU of x0, such that

f(x) ≥ f(x0) for all x ∈ U.

48 The Derivative

The next result is commonly known as theFirst Derivative Test.

Task 123 Supposef : [a, b] → R has either a relative maximum or arelative minimum atx0 ∈ (a, b). If f is differentiable atx0, thenf ′(x0) = 0.

Task 124 Supposef : [a, b] → R is continuous on[a, b] and differen-tiable on(a, b).If f(a) = f(b) = 0, then there exists ac ∈ (a, b) with f ′(c) = 0.

A much more useful version of Task124is known as theMean Value The-orem:

Figure 10: The Mean Value Theorem

Task 125 Supposef : [a, b] → R is continuous on[a, b] and differen-tiable on(a, b).

6.3 The Mean-Value Theorem and its Applications 49

Then there exists ac ∈ (a, b) such that

f ′(c) =f(b)− f(a)

b− a.

See Figure10on page48.

Do not confuse the Mean Value Theorem with the Intermediate Value The-orem!

Nearly all properties of differentiable functions follow from the Mean ValueTheorem. The exercises and tasks below are such examples of straightfor-ward applications of the Mean-Value Theorem.

Exercise 126 Let f : [a, b] → R be continuous on[a, b] and differen-tiable on(a, b).If f ′(x) > 0 for all x ∈ (a, b), thenf is strictly increasing.

Exercise 127 Let f : [a, b] → R be continuous on[a, b] and differen-tiable on(a, b).If f ′(x) = 0 for all x ∈ (a, b), thenf is constant on[a, b].

A function f : D → R is called INJECTIVE (or 1–1), if x 6= y impliesf(x) 6= f(y) for all x, y ∈ D.

Exercise 128 Let f : [a, b] → R be continuous on[a, b] and differen-tiable on(a, b).If f ′(x) 6= 0 for all x ∈ (a, b), thenf is injective.

50 The Derivative

Task 129 Let f : [a, b] → R be differentiable on[a, b] such thatf ′(x) 6= 0 for all x ∈ [a, b].Thenf is injective; its inversef−1 is differentiable onf([a, b]). Moreover,settingy = f(x), we have (

f−1)′

(y) =1

f ′(x).

6.4 The Derivative and the Intermediate Value Propertyopt

We say that a functionf : [a, b] → R has the INTERMEDIATE VALUE

PROPERTYif the following holds: Letx1, x2 ∈ [a, b], and let

y ∈ (f(x1), f(x2)).

Then there is anx ∈ (x1, x2) satisfyingf(x) = y.

Recall that we saw earlier that every continuous function has the intermedi-ate value property, see Task109.

On the other hand, not every function with the intermediate value propertyis continuous:

Optional Task 130 Let f : [−1, 1] → R be defined by

f(x) =

sin(

1x

), if x 6= 0, x ∈ R

0, if x = 0

Show thatf has the intermediate value property on the interval[−1, 1].See Figure5 on page25.

The rest of this section will establish the surprising fact that derivatives havethe intermediate value property, even though they are not necessarily con-tinuous (see Task115).

6.4 The Derivative and the Intermediate Value Propertyopt 51

Optional Task 131 Let f : [a, b] → R be differentiable on[a, b].If f ′(x) 6= 0 for all x ∈ (a, b), then eitherf ′(x) ≥ 0 for all x ∈ [a, b] orf ′(x) ≤ 0 for all x ∈ [a, b].

Optional Task 132 Let f : [a, b] → R be differentiable on[a, b].Thenf ′ : [a, b] → R has the intermediate value property on[a, b].

52 The Derivative

7 The Integral [Under Construction]

Throughout this chapter all functions are assumed to be bounded.


A finite setP = x0, x1, x2, . . . , xn is called aPARTITION of the interval[a, b], if

a = x0 < x1 < x2 < · · · < xn = b.

Figure 11: A partition of the interval[a, b].

Let a functionf : [a, b] → R and a partitionP of the interval[a, b] be given.Let i ∈ 1, 2, 3, . . . , n. We define

mi(f) := inff(x) | x ∈ [xi−1, xi],

andMi(f) := supf(x) | x ∈ [xi−1, xi].

TheLOWER RIEMANN SUM L(f, P ) is defined as

L(f, P ) :=n∑

i=1

mi(f)(xi − xi−1).


TheUPPERRIEMANN SUM U(f, P ) is defined as

U(f, P ) :=n∑

i=1

Mi(f)(xi − xi−1).



Figure 12: A lower Riemann sum with a partition consisting of 20 equally-spaced points.

Figure 13: An upper Riemann sum with a partition consisting of 12 equally-spaced points.

TheLOWER RIEMANN INTEGRAL of f on the interval[a, b] is defined as

L∫ b

a

f(x) dx := supL(f, P ) | P is a partition of[a, b].

TheUPPERRIEMANN INTEGRAL of f on the interval[a, b] is defined as

U∫ b

a

f(x) dx := infU(f, P ) | P is a partition of[a, b].

LetP andQ be two partitions of the interval[a, b]. We say that the partitionQ is FINER than the partitionP if P ⊆ Q. In this situation, we also callPCOARSERthanQ.

7.1 Definition and Examples 55

Task 133 Let a functionf : [a, b] → R and two partitionsP andQ ofthe interval[a, b] be given. Assume thatQ is finer thanP . Then

L(f, P ) ≤ L(f,Q) ≤ U(f,Q) ≤ U(f, P ).

Note that Task133implies that

L∫ b

a

f(x) dx ≤ U∫ b

a

f(x) dx.

We are finally in a position to define the concept of integrability! We willsay that a functionf : [a, b] → R is RIEMANN INTEGRABLE on the interval[a, b], if

L∫ b

a

f(x) dx = U∫ b

a

f(x) dx.

Their common value is then called the RIEMANN INTEGRAL of f on theinterval[a, b] and denoted by ∫ b

a

f(x) dx.

Task 134 Use the definition above to computeL∫ 1

0

x dx and

U∫ 1

0

x dx. Is the function Riemann integrable on[0, 1]?

Task 135 Let f : [0, 1] → R be defined by

f(x) =

1, if x ∈ Q0, if x ∈ R \Q

Use the definitions above to computeL∫ 1

0

f(x) dx andU∫ 1

0

f(x) dx. Is the

function Riemann integrable on[0, 1]?


Task 136 A function f : [a, b] → R is Riemann integrable if and onlyif for everyε > 0 there is a partitionP of [a, b] such that

U(f, P )− L(f, P ) < ε.

Given a partitionP , we define itsMESH WIDTH ω(P ) as

ω(P ) := maxxi − xi−1 | i = 1, 2, . . . , n.

Task 137 A function f : [a, b] → R is Riemann integrable if and onlyif for everyε > 0 there is aδ > 0 such that forall partitionsP of [a, b] withmesh widthω(P ) < δ

U(f, P )− L(f, P ) < ε.

Task 138 If f : [a, b] → R is continuous on[a, b], thenf is Riemannintegrable on[a, b].

Task 139 If f : [a, b] → R is increasing (or decreasing) on[a, b], thenf is Riemann integrable on[a, b].

7.2 Arithmetic of Integrals

Task 140 Let f : [a, b] → R be Riemann integrable on[a, b]. Then forall λ ∈ R, the functionλf is also Riemann integrable on[a, b], and∫ b

a

λf(x) dx = λ

∫ b

a

f(x) dx.

7.3 The Fundamental Theorem of Calculus 57

Task 141 Let f, g : [a, b] → R be Riemann integrable on[a, b]. Thenf + g is also Riemann integrable on[a, b], and∫ b

a

f(x) + g(x) dx =

∫ b

a

f(x) dx+

∫ b

a

g(x) dx.

Task 142 Let f : [a, c] → R be a function anda < b < c. Thenf isRiemann integrable on[a, c] if and only if f is Riemann integrable on both[a, b] and[b, c]. In this case∫ c

a

f(x) dx =

∫ b

a

f(x) dx+

∫ c

b

f(x) dx.

Task 143 Suppose the functionf : [a, b] → R is bounded above byM ∈ R: f(x) ≤M for all x ∈ [a, b]. Then∫ b

a

f(x) dx ≤M · (b− a).

7.3 The Fundamental Theorem of Calculus

Task 144 Let f : [a, b] → R be continuous. Then there is a numberc ∈ [a, b] such that

f(c) =1

b− a

∫ b

a

f(x) dx.


Task 145 Let f : [a, b] → R be bounded and Riemann integrable on[a, b]. Let

F (x) =

∫ x

a

f(τ) dτ.

ThenF : [a, b] → R is continuous on[a, b].

Task 146 Let f : [a, b] → R be continuous on[a, b]. Let

F (x) =

∫ x

a

f(τ) dτ.

ThenF : [a, b] → R is differentiable on[a, b], and

F ′(x) = f(x).

The next result is theFundamental Theorem of Calculus.

Task 147 Supposef : [a, b] → R is Riemann integrable on[a, b], andsupposeF : [a, b] → R is an ”anti-derivative” off(x), i.e.,F satisfies:

1. F is continuous on[a, b] and differentiable on(a, b),

2. F ′(x) = f(x) for all x ∈ [a, b].

Then ∫ b

a

f(τ) dτ = F (b)− F (a).

A Hints and Remarks

Task 1 Assume that√

2 =a

bwith a, b ∈ N andgcd(a, b) = 1. Thus

2b2 = a2, soa is divisible by2. Show thatb is also divisible by2.

Task12 Assume that the sequence converges to botha andbwith a 6= b.Derive a contradiction, since far out sequence terms must be close to bothaandb.

Task 21 Once we know that the sequence converges, we can find its limit asfollows: LetL = lim

n→∞an. Then lim

n→∞an+1 = L as well, and lim

n→∞

√3an =

√3L.

ThusL satisfies the equationL =√

3L. SinceL 6= 0, we conclude that the limitof the sequence must be3.

Task 32 The following definition might be useful: We say that the se-quence(an) has aPEAK atn0 if

an0 ≥ an for all n ≥ n0.

The result stated in the text has a beautiful generalization due to Frank P. Ramsey(1903–1930), which unfortunately requires some notation: Given an infinite subsetM of N, we denote the set of doubletons fromM by

P(2)(M) := m,n | m,n ∈ M andm < n .

Ramsey’s Theorem.LetA be an arbitrary subset ofP(2)(N). Then there is aninfinite subsetM of N such that either

P(2)(M) ⊆ A, or P(2)(M) ∩ A = ∅.

The result in Task32 is an easy consequence of Ramsey’s Theorem: Set

A = m,n | m < n andam ≤ an.

Task 36 An object is well-defined if it exists and is uniquely deter-mined.

60

Task 60 Show first that the set

Fq =

x ∈ (0, 1] | f(x) ≥ 1

q

is finite for all q ∈ N.

This implies thatFq has no accumulation points.

Task 62 Show the following: Suppose there are two sequences(xn) and(yn), both converging tox0, such that(f(xn)) converges toL and(f(yn))converges toM . If 2. holds, thenL = M .

Task 69 Show first that the set

Dn := y ∈ (a, b) | (U(y)− L(y)) > 1/n

is finite for alln ∈ N.

Task 71 Defineϕ : N → 2N by ϕ(n) = 2n.

Task 72 Defineϕ : Z → N by

ϕ(z) =

2z, if z ≥ 0

−(2z + 1), if z < 0

Task 73 First show that the positive rational numbersQ+ are countable.Start by observing that

Q+ = 1

1,

2

1,1

2,

3

1,2

2,1

3,

4

1,3

2,2

3,1

4, . . .

Task 74 Suppose there is a bijectionϕ : A → P(A). Note that fora ∈ A, ϕ(a) is an element ofP(A), and thus a subset ofA. Consider the set

B = a ∈ A | a 6∈ ϕ(a).

Let b ∈ A be such thatϕ(b) = B. Is b ∈ B? Isb 6∈ B?

61

Task 75 Defineϕ : R → (0, 1) by

ϕ(x) =1

2+

1

πarctanx.

Task 76 This is more obnoxious than you might think. Defineϕ :(0, 1) → [0, 1] in the following way:ϕ(1

2) = 0, ϕ(1

3) = 1, ϕ( 1

n) = 1

n−2

for n ∈ 4, 5, 6, 7, . . ., andϕ(x) = x otherwise.

Task 78 Supposex ∈ [0, 1] has two distinct binary expansions:

x =∞∑

n=1

δn2n

andx =∞∑

n=1

εn

2n.

Then without loss of generality there is a uniquek ∈ N such thatδn = εn

for all n < k andδk = 0, while εk = 1. This implies that

x =ε1

21+ε2

22+ · · ·+ εk−1

2k−1+

1

2k.

ConsequentlyB consists of all elements in(0, 1) with a terminating binaryexpansion, and is thus indeed countable:

B =

1

2,

1

4,3

4,

1

8,3

8,5

8,7

8, · · ·

Task 79 Let B = bn | n ∈ N. Pick another countable setC ⊂ Asuch thatB ∩ C = ∅. (How can one do that?) LetC = cn | n ∈ N.SinceA ∼ P(N), there is a bijectionϕ : A → P(N). Define the bijectionψ : A\B → P(N) as follows: OnC setψ(c2n−1) = ϕ(bn),ψ(c2n) = ϕ(cn);for x ∈ A \ (B ∪ C) defineψ(x) = ϕ(x).

Task 80 Define a mapϕ : [0, 1] → P(N), which “essentially” is thedesired bijection:

ϕ(x) = ϕ

(∞∑

n=1

εn

2n

)= n ∈ N | εn = 1

62

For instance,ϕ

(5

8

)= 1, 3, andϕ

(1

3

)= 2N. Vice versa, the set of

odd integers, for instance, corresponds to2

3:

ϕ−1 (1, 3, 5, 7, 9, . . .) =1

2+

1

23+

1

25+ . . . =

2

3

Every x ∈ [0, 1] defines one (or at most 2) subsets ofN via ϕ, and viceversa, every subset ofN defines a real number in[0, 1] via ϕ−1.

Task 81 Let P be a perfect set.P cannot be finite, so supposeP iscountable, sayP = p1, p2, p3, . . .. Find a sequence of closed boundedintervalsIn such that (1)In ∩ P 6= ∅ (2) In ⊆ In−1 (3) pn 6∈ In+1 for alln ∈ N and (4) the lengths of the intervalsIn converge to0. Next show that∞⋂

n=1

In is not empty and that its elements are accumulation points ofP (and

therefore inP ). This leads to a contradiction.

Task 82 1. Supposex 6∈ C. Then there is ann ∈ N such thatx 6∈Cn. Note that the complement ofCn consists of a finite number of openintervals; therefore there is a neighborhoodV of x such thatV ∩ Cn = ∅.ConsequentlyV ∩ C = ∅, and thusx is not an accumulation point ofC.

2. First show that for alln ∈ N the “endpoints” of the closed intervalsCn

consists of are members ofC. In particular,C 6= ∅. Now letx ∈ C and letε > 0. By choosingn sufficiently large we can ensure that the neigborhood(x− ε, x+ ε) contains one of the closed intervalsCn is comprised of; hence(x − ε, x + ε) contains the two endpoints of this interval, at least one ofwhich must be different fromx. This shows thatx is an accumulation pointof C.

Task 92 In the late 1890s Rene-Louis Baire (1874–1932) proved that thereare no functions on the real line that are continuous at all rational numbers anddiscontinuous at all irrational numbers.

Task 100 What does it mean iff fails to be uniformly continuous?Along the way, you probably want to use the Bolzano-Weierstrass Theorem(Task33on page17).

63

Task 106 Note that every continuous function on a closed interval isautomatically uniformly continuous, see Task100.

Task 122 You might want to look at the statement of Task97.

Task 123 This only works forx0 ∈ (a, b), not if x0 is one of the end-pointsa or b.

Task 137 The proof of this result is rather technical. Note that giventwo partitionsP andQ, the partitionP ∪Q is finer than bothP andQ.

64

B Greek Letters

α alpha β beta γ gamma δ deltaε, ε epsilon ζ zeta η eta θ, ϑ thetaι iota κ kappa λ lambda µ muν nu ξ xi o omikron π piρ, % rho σ sigma τ tau υ upsilonφ, ϕ phi χ chi ψ psi ω omega

Γ Gamma ∆ Delta Θ Theta Λ LambdaΞ Xi Π Pi Σ Sigma Υ UpsilonΦ Phi Ψ Psi Ω Omega

66

C References for Additional Reading

One can start an Analysis course by “constructing” the real numbers froman axiomatic system for the set of natural numbers. The book below con-tains such a construction; it then continues to discuss complex numbers,quaternions, and division algebras in general.

Heinz-Dieter Ebbinghaus, John H. Ewing (Editors): Numbers.Springer-Verlag, Reprint edition 1995.

The “Discovery Learning Project” is designed to disseminate knowledgeabout the teaching legacy of R.L. Moore:

“The Legacy of R.L. Moore”. Retrieved March 28, 2003.http://www.discovery.utexas.edu/rlm/index.html

“Mathnerds” is a Moore-style web site for students. It also features quite afew theorem sequences similar to the one in this text:

“MathNerds.” Retrieved March 28, 2003.http://www.mathnerds.com/

The reference below is a wonderful reference if you are interested in thehistory of mathematics and mathematicians:

”The MacTutor History of Mathematics archive”. RetrievedMarch 28, 2003.http://www-groups.dcs.st-and.ac.uk/ history/index.html

c©2002–2004. Last edits: December 1, 2004

http://www.discovery.utexas.edu/rlm/index.html

http://www.mathnerds.com/

http://www-groups.dcs.st-and.ac.uk/~history/index.html

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