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.