Math 151
Section 2.5
Continuity
Continuous A function f ( x ) is said to be continuous at x = a if lim f ( x) = f ( a) .
x!a
We say f ( x ) is continuous from the left if lim" f ( x) = f ( a) . We say f ( x ) is continuous from the
right if lim+ f ( x) = f ( a) .
x!a
x!a
Example: Use the graph of f ( x ) to locate all discontinuities of f ( x ) . For each, identify the limit
from the left and the limit from the right and state whether it is continuous from the right, left, or
neither.
Math 151
Example: Explain why the following functions are not continuous at the indicated value of x.
A. f ( x) =
!1
(1! x)
2
,x =1
#% x 2 ! 2x !8
%
B. f ( x) = %
$ x!4
%%
%&3
if x " 4
if x = 4
#2x +1
%
%
%
%3x
Example: Identify all points of discontinuity for f ( x) = %$
%
4
%
%
%
2
%
&x + 2
Example: For what value(s) of c is g ( x) continuous, if any?
#
%x 2 ! c2
g ( x) = %
$
%
%
&cx + 20
if x < 4
if x " 4
if x ! "1
if "1< x <1
if x = 1
if x >1
Math 151
Intermediate Value Theorem If f ( x ) is continuous on the interval [a, b] and N is any number
strictly between f ( a) and f (b) , then there exists a number c in (a, b) such that f (c) = N.
Example: If g ( x) = x 5 ! 2x 3 + x 2 + 2 , show that there is a number c such that g (c) = !1 .
Example: Show that h( x) = x 5 ! 2x 4 ! x ! 3 has a root on the interval (2, 3).