2.7: Continuity and the
Intermediate Value Theorem
Objectives:
•Define and explore
properties of continuity
•Introduce Intermediate
Value Theorem
©2002 Roy L. Gover (roygover@att.net)
Definition
f(x) is continuous at x=c
if and only if there are
no holes, jumps, skips
or gaps in the graph of
f(x) at c.
Examples
Continuous Functions
Examples
Discontinuous Functions
Infinite
discontinuity
(nonJump
Discontinuity
(nonRemovable
discontinuity
removable)
removable)
Definition
f(x) is continuous at x=c if
and only if:
1. f (c) is defined …and
f ( x ) exists …and
2. lim
x c
f ( x )  f (c )
3. lim
x c
Examples
Discontinuous
at x=2
because f(2)
is not defined
x=2
Examples
Discontinuous
at x=2
because,
although f(2)
is defined,
lim f ( x)  f (2)
x 2
x=2
Definition
f(x) is continuous on the
open interval (a,b) if and
only if f(x) is continuous at
every point in the interval.
Try This
Find the values of x (if any)
where f is not continuous. Is
the discontinuity removable?
f ( x) 
0, for x  0
x , for x  0
2
Continuous for all
x
Try This
Find the values of x (if any)
where f is not continuous. Is
the discontinuity removable?
1
f ( x) 
x
Discontinuous at x=o, not
removable
Example
Find the values of k, if possible,
that will make the function
continuous.
f ( x) 
kx , for x  2
2
2 x  k , for x  2
Definition
f(x) is continuous on the
closed interval [a,b] iff it is
continuous on (a,b) and
continuous from the right at
a and continuous from the
left at b.
Example
f(x)
f(x)isis
continuous
on
a
f(x)
is
continuous
f(x)
(continuous
a,b
)
from the right
from
the
left
at a
b
at b
f(x) is continuous on [a,b]
Graphing calculators can make non-continuous
functions appear continuous.
Graph:
y  floor  x 
CATALOG
F
floor(
Note resolution.
The calculator “connects the dots”
which covers up the discontinuities. 
Graphing calculators can make non-continuous
functions appear continuous.
Graph:
y  floor  x 
CATALOG
F
floor(
If we change the plot style
to “dot” and the resolution
to 1, then we get a graph
that is closer to the
correct floor graph.
The open and closed circles do not
show, but weGRAPH
can see the
discontinuities.
p
Intermediate Value Theorem
If f is continuous on [a,b]
and k is a number
between f(a) & f(b), then
there exists a number c
between a & b such that
f(c ) =k.
Intermediate Value Theorem
f(a)
k
f(b)
a
c
b
Intermediate Value Theorem
•an existence theorem; it
guarantees a number exists
but doesn’t give a method
for finding the number.
•it says that a continuous
function never takes on 2
values without taking on all
the values between.
Example
Kaley was 20 inches long when
born. Let’s say that she will be
30 inches long when 15 months
old. Since growth is continuous,
there was a time between birth
and 15 months when she was 25
inches long.
Try This
Use the Intermediate Value
Theorem to show that
f ( x)  x
3
has a zero in the interval
[-1,1].
f ( 1)  1
Solution
3
f (1)  1
f ( x)  x
therefore, by the
Intermediate
Value Theorem,
there must be a
f (c)=0 where
1  c  1