The Intermediate Value Theorem

Introduction

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

and , then for all values between and , there exists between and such that 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:

Domain: (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 and , there exists such that 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:

Domain: (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 and , there exists such that equals that value.

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