Calculus Section 2.5
Continuity
Homework: Pages 152 # 1 - 29 odd and #28
Objective: SWBAT determine if a function is continuous.
1. Definition of Continuity - a curve that is unbroken without gaps, jumps, or holes.
2. Definition 2.5.1
A function f is said to be continuous at a point c if the following
conditions are satisfied:
a. f (c ) is defined
f ( x) exists
b. lim
x c
f ( x )  f (c )
c. lim
x c
3. Discontinuous at c - if one or more of the conditions in definition 2.5.1 fails. If the definition
fails then c is called a point of discontinuity of f.
4. Removable Discontinuity
If at some point x  c the lim f ( x ) exists but the f (c ) is not equal to the lim f ( x ) then
x c
the function is said to have a removable discontinuity at x  c .
x c
5. Jump Discontinuity
When the one sided limits at a point are not equal.
6. Determine if the given functions are continuous or discontinuous. If the function is
discontinuous determine if the discontinuity is removable or a jump.
x2  4
x2
a.
f ( x) 
b.
 x2  4
,

f ( x)   x  2
3,

c.
 x2  4
,

f ( x)   x  2
4,

x2
x2
x2
x2
a. All polynomials are continuous functions.
b. A rational function is continuous at every point where the denominator is nonzero, and
has a discontinuity at the points where the denominator is zero.
A function f is said to be continuous on a closed interval  a, b if the following conditions
are satisfied:
a. f is continuous on  a, b 
b. f is continuous from the right at a
c. f is continuous from the left at b
7. Properties of Continuous Functions
Theorem 2.4.3 If the functions f and g are continuous at c, then
a. f  g is continuous at c;
b. f  g is continuous at c;
c. f  g is continuous at c;
f
d. is continuous at c if g (c)  0 and is discontinuous at c if g (c)  0 .
g
8. For what values of x is there a discontinuity in the graph of h( x) 
9. Show that f ( x)  x  3  2 is continuous everywhere.
x2  9
?
x2  5x  6
10. Suppose lim f ( x)  1, lim f ( x)  1, and f (3) is not defined. Which of the following
x 3
x 3
statements is (are) true?
a. lim f ( x)  1
x 3
b. f is continuous everywhere except at x  3
c. f has a removable discontinuity at x  3
11. Example 7
 x2  x
,

If f ( x)   2 x
 f (0)  k

x0
and if f is continuous at x  0 , then k 
12.
1  x,
 2
2 x  2,

Given f ( x)   x  2,
1,

2 x  4,
1  x  0
0  x 1
1 x  2
x2
2 x3
a. Find lim f ( x )
x2
b. The function has a removable discontinuity at?
c. On what intervals is the function continuous?
The function has a jump discontinuity at?
13. Continuity of Composition (Theorem 2.5.5)
If lim g ( x)  L and if the function f ( x) is continuous at L, then lim f ( g ( x))  f ( L) . In other
x c

words lim f ( g ( x))  f lim g ( x)
x c
x c

x c
A limit symbol can be moved through a function sign provided the limit of the
expression inside the function sign exists and the function is continuous at this limit.
14. Previously we saw that f ( x)  x  3  2 is continuous everywhere, so Theorem 2.5.5
implies that
lim g (c)  lim g ( x)
x c
x c
15.
a. If the function g ( x ) is continuous at the point c and the function f ( x) is
continuous at the point g (c) then the composition f ( g (c)) is continuous.
b. If the function g ( x ) is continuous everywhere and the function f ( x) is
continuous everywhere, then the composition f ( g ( x)) is continuous everywhere.
16. The absolute value of a continuous function is continuous.
17. The Intermediate Value Theorem - If f is continuous on a closed interval  a, b and
k is any number between f (a ) and f (b ) inclusive, then there is at least one number x in the
interval  a, b such that f ( x)  k .
18. If f is continuous on  a, b and if f (a ) and f (b ) are nonzero and have opposite signs,
then there is at least one solution of the equation f ( x)  0 in the interval  a, b  .
19. Show that x3  x  1  0 has at least one solution in the interval 1, 2  , use an interval
to bracket the answer.
x
f ( x)
x
f ( x)
1
-1
1.1
-.77
1.3
1.31
-.103 -.062
1.2
-.47
1.3
-.10
1.4
.34
1.5
.88
1.6
1.50
1.7
2.21
1.8
3.03
1.9
3.96
2.0
5
1.32
-.020
1.33
.023
1.34
.066
1.35
.110
1.36
.155
1.37
.201
1.38
.248
1.38
.296
1.39
.344