Chapter 3.2 Notes
1
(c) Epstein, 2014
3.2 Continuity
f ( x) is continuous at a number a if lim f ( x) = f (a)
x a
5
f ( x)
Example:
Where is the graph discontinuous?
-5
5
x
f ( x) is continuous from the right at a number a if
lim f ( x ) = f ( a )
x a+
f ( x) is continuous from the left at a number a if
lim f ( x ) = f ( a )
x  a-
So to check for continuity, do the following:
1. Make sure f  x  is defined at x  a
2. Check that lim f ( x) exists
x a
3. Finally, check that lim f ( x) = f (a)
x a
Chapter 3.2 Notes
2
(c) Epstein, 2014
Example: Explain why each function is discontinuous at the given point.
x 2 -1
a) a = -1 , f ( x) =
x +1
b) a = 4 ,
ìï x 2 - 2 x - 8
ïï
if x ¹ 4
f ( x) = í x - 4
ïï
if x = 4
ïî3
A function f ( x) is continuous on an interval if it is continuous at every
number in the interval. At the endpoints it is understood that the function
is left or right continuous.
Continuity Rules
If f and g are continuous at a and c is a constant, then the following
functions are also continuous at a
f
f g
cf
fg
(provided that g (a) ¹ 0 )
g
The following are continuous wherever they are defined:
1. Polynomials
2. Rational functions
3. Power functions
4. Trigonometric functions
5. Exponential functions of the form a x , a  0 , and a  1
6. Logarithmic function of the form log a x , a  0 , and a  1
Chapter 3.2 Notes
3
(c) Epstein, 2014
Example: Find where the given functions are discontinuous.
ìï2 x + 1 if x £ -1
ïï
a) g ( x ) = í3 x
if -1 < x < 1
ïï
ïïî2 x + 1 if x ³ 1
b) h(t ) =
t +1
4t 2 -1
ìï -x
ïï
c) h( x) = ïí1
ïï
ïïî x
if x £ 0
if 0 < x < 1
if x ³ 1
Chapter 3.2 Notes
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(c) Epstein, 2014
Example: Find the value or values of c that make f continuous
ì
ï
x 2 - c 2 if x < 4
ï
f ( x) = í
ï
ï
îcx + 20 if x ³ 4
If g  x  is continuous at x  a with g  a   L , and f  x  is continuous at
x  L , then  f  g  x  is continuous at x  a . That is,
lim ( f  g )( x) = lim f éë g ( x)ùû = f éêlim g ( x)ùú = f éë g (a)ùû = f ( L)
x a
x a
ë x a
û
Example: Find where the given functions are discontinuous
a) F ( x) = 2 x + 25 - x 2
b) g (t ) =
1
t + t2 - 4
c) G ( x) = x - x 2
d) H ( x ) =
x-2
5+ x
Chapter 3.2 Notes
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(c) Epstein, 2014
A removable discontinuity at a is a discontinuity that can be removed by
redefining the function at a.
Example:
Which of the following functions f has a removable discontinuity at a? If
it is removable, redefine f so that the function is continuous on  .
a) a = -2,
b) a = 7,
x2 - 2 x - 8
f ( x) =
x+2
f ( x) =
x-7
x-7
Chapter 3.2 Notes
c) a = -4,
d) a = 9,
6
x3 + 64
f ( x) =
x+4
f ( x) =
3- x
9- x
(c) Epstein, 2014