DEFINITION Continuity at a Point f ( x ) is defined on an open interval containing
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31
2.3 Quiz
lim ?4x
x
x x
lim / lim / lim , lim 0
x c x c x c x cf x g x f x g x g x
1 1
5 5
DEFINITION Continuity at a Point f (x) is defined on an open interval containing x = c. If , then f is continuous at x = c .
If the limit does not exist, or if it exists but is not equal to f (c), we say that f has a discontinuity at x = c.
( )a b
Three conditions must hold:
For a function to be continuous at a point...
1. is defined
2. lim exists
3. limx c
x c
f c
f x
f x f c
is continuous at f x c
A function f (x) may be continuous at some points and discontinuous at others. If f (x) is continuous at all points in an interval I, then f (x) is said to be continuous on I. If I is an interval [a, b] or [a, b) that includes a as a left endpoint, we require that
lim .x a
f x f a
Similarly, we require that
if I includes b as a right endpoint. If f (x) is continuous at all points in its domain, then f (x) is simply called continuous.
lim ,x b
f x f b
( )a b
is the open interval , ,
& is continuous on .
I a b
f I
[ ]
limx cf x
exists but is not equal to f (c)
F has a removable discontinuity at x = c.
, 25, 2{ f x xx
f x
A “worse” type of discontinuity is a jump discontinuity, which occurs if the one-sided limits and exist but are not equal. Below are two functions with jump discontinuities at c = 2. Unlike the removable case, we cannot make f (x) continuous by redefining f (c).
limx c
f x
limx c
f x
DEFINITION One-Sided Continuity A function f (x) is called:
•Left-continuous at x = c if •Right-continuous at x = c if
limx c
f x f c
limx c
f x f c
Piecewise-Defined Function Discuss the continuity of
At x = 1, the one-sided limits exist but are not equal:
has a jump discontinuity and is right-continuous at 1.F x x
At x = 3, the left- and right-hand limits exist and both are equal to F (3), so F (x) is continuous at x = 3:
f (x) has an infinite discontinuity at x = c if one or both of the one-sided limits is infinite. Notice that x = 2 does not belong to the domain of the function in cases (A) and (B).
Some functions have more “severe” types of discontinuity. For example, oscillates infinitely often between +1
and −1 as x → 0. Neither the left- nor the right-hand limit exists at x = 0, so this discontinuity is not a jump discontinuity.
1sinf xx
It is easy to evaluate a limit when the function in question is known to be continuous.
limx cf x f c
.
3
lim sin3
2x
x
1
3 1/
6
3im
2
1l
5
x
x x
End Day 1 InstructionBegin
.
Building Continuous Functions
THEOREM 1 Basic Laws of Continuity If f (x) and g (x) are continuous at x = c, then the following functions are also continuous at x = c:
(iv) f (x)/g (x) if g (c) 0
(i) f (x) + g (x) and f (x) – g (x)
(ii) kf(x) for any constant k
(iii) f (x) g (x)
.
Building Continuous Functions
THEOREM 2 Continuity of Polynomial and Rational Functions Let P(x) and Q(x) be polynomials. Then:
(i) P(x) is continuous on the real line.
(ii) P(x)/Q(x) is continuous on its domain.
Building Continuous FunctionsTHEOREM 3 Continuity of Some Basic Functions
(iv) is continuous for (for ).
(i) is continuous on its domain for n a natural #.
(ii) are continuous on the real line.
(iii) is continuous on the real line (for ).
1/ ny x
sin and cosy x y x xy b
logby x 0x 0, 1b b
0, 1b b
1/The domain of is the real line if is odd
and [0, ) if is even.
ny x n
n
x 2x
y 0
1
2
3
1
2
4
8
The base of a logarithmic
function has to be positive.
.
As the graphs suggest, these functions are continuous on their domains.
Because sin x and cos x are continuous, Continuity Law (iv) for Quotients implies that the other standard trigonometric functions are continuous on their domains.
(i.e.) there are no jump discontinuities
.
As the graphs suggest, these functions are continuous on their domains.
Because sin x and cos x are continuous, Continuity Law (iv) for Quotients implies that the other standard trigonometric functions are continuous on their domains.
sintan
cos
xx
x
tanf x x
Building Continuous Functions
THEOREM 4 Continuity of Composite Functions
If is continuous at , and is continuous
at , then the composite funct
is continuous
i
at
n
o
.
g x c f
x g
F x f g x c
c
x
1/32Is 9 continuous?F x x
is the composite of two continuous functionsF
Yes
Building Continuous Functions
THEOREM 4 Continuity of Composite Functions
If is continuous at , and is continuous
at , then the composite funct
is continuous
i
at
n
o
.
g x c f
x g
F x f g x c
c
x
1Is cos continuous?F x x
1 is continuous 0.g x x x
cos is continuous .f x x
, but is continuous
at all points in its domain.
No F
Building Continuous Functions
THEOREM 4 Continuity of Composite Functions
If is continuous at , and is continuous
at , then the composite funct
is continuous
i
at
n
o
.
g x c f
x g
F x f g x c
c
x
sinIs 2 continuous?xF x
is the composite of two continuous functionsF
Yes
Building Continuous Functions
THEOREM 4 Continuity of Composite Functions
If is continuous at , and is continuous
at , then the composite funct
is continuous
i
at
n
o
.
g x c f
x g
F x f g x c
c
x
2 cos 2 9sin
8
xxF x
x
A function constructed of basic functions
using operatio
Elementary Function
Basic Func
ns and compositions.
A function that's continuous on its do
ti
m n
on
ai .
Continuous : 8x x
2 1.7 1
The Greatest Integer Function
, where is the unique integer such that
1.
x n n
n x n
1.7 ? 2
f x x
2
limx
x
We cannot use susbstitution because
is not continuous.f x x
DNE
Complete
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