The Intermediate Value Theorem The Intermediate Value Theorem

Introduction

The Intermediate Value Theorem states that if a function is continuous between

a
and
b
, then for all values between
f(a)
and
f(b)
, there exists
c
between
a
and
b
such that
f(c)
equals that value.

In mathematical notation:
If
f
is continuous on
\lbrack a, b \rbrack
, then
\forall y \in \lbrack f(a), f(b) \rbrack, \exists c \in \lbrack a, b \rbrack
such that
f(c) = y
.


Here is an example of the Intermediate Value Theorem:
If
f(-5)=2
and
f(9)=7
, and
f
is continuous between
-5
and
9
,
then we know that there exists a value
c
between
-5
and
9
such that
f(c)=5, f(c)=2.1, f(c)=\pi,
etc.

For any value between
2
and
7
, there exists a value
c
such that
f(c)
equals that value.

Examples

Example 1:

f(x)=x^2

Domain:
[-1,2]
(Chosen arbitrarily)

The Intermediate Value Theorem states that since
f(x)
is continuous on
[-1,2]

f(-1)=1

f(2)=4

Therefore, for any value between
1
and
4
, there exists
c
such that
f(c)
equals that value.

In other words, for any value between the blue lines, there exists a spot on the red line at that height.
Example 2:
f(x)=x^3

Domain:
[-2,2]
(Chosen arbitrarily)

The Intermediate Value Theorem states that since
f(x)
is continuous on
[-2,2]

f(-2)=-8

f(2)=8

Therefore, for any value between
-8
and
8
, there exists
c
such that
f(c)
equals that value.

In other words, for any value between the blue lines, there exists a spot on the red line at that height.