##
The Intermediate Value Theorem

Your teacher probably told you that you can draw the graph of a
continuous function without lifting your pencil off the paper.
This is made precise by the following result:

**Intermediate Value Theorem.** Let *f* (*x*) be a continuous
function on the interval [*a*, *b*]. If
*d* [*f* (*a*), *f* (*b*)], then
there is a
*c* [*a*, *b*] such that *f* (*c*) = *d*.

In the case where *f* (*a*) > *f* (*b*),
[*f* (*a*), *f* (*b*)] is meant to be the
same as
[*f* (*b*), *f* (*a*)]. Another way to state the Intermediate
Value Theorem is to say that the image of a closed interval under
a continuous function is a closed interval. We will present an
outline of the proof of the Intermediate Value Theorem on the
next page.

Here is a classical consequence of the Intermediate Value Theorem:

**Example.** Every polynomial of odd degree has at least one
real root.

We want to show that if
*P*(*x*) = *a*_{n}*x*^{n} + *a*_{n - 1}*x*^{n - 1} + ^{ ... } + *a*_{1}*x* + *a*_{0} is a polynomial with *n* odd and *a*_{n} 0,
then there is a real number *c*, such that *P*(*c*) = 0.

First let me remind you that it follows from the results in
previous pages that every polynomial is continuous on the real
line. There you also learned that

Consequently for | *x*| large enough, *P*(*x*) and *a*_{n}*x*^{n} have
the same sign. But *a*_{n}*x*^{n} has opposite signs for positive *x*
and negative *x*. Thus it follows that if *a*_{n} > 0, there are real
numbers *x*_{0} < *x*_{1} such that *P*(*x*_{0}) < 0 and *P*(*x*_{1}) > 0. Similarly
if *a*_{n} < 0, we can find *x*_{0} < *x*_{1} such that *P*(*x*_{0}) > 0 and
*P*(*x*_{1}) < 0. In either case, it now follows directly from the
Intermediate Value Theorem that (for *d* = 0) there is a real
number
*c* [*x*_{0}, *x*_{1}] with *P*(*c*) = 0.

The natural question arises whether every function which
satisfies the conclusion of the Intermediate Value Theorem must
be continuous. Unfortunately, the answer is no and
counterexamples are quite messy. The easiest counterexample is
the function

*f* (

*x*) =

As you found out on an earlier page, this function fails to be
continuous at *x* = 0. On the other hand, it is not too hard to see
that *f* (*x*) has the "Intermediate Value Property" even on closed
intervals containing *x* = 0.

**
**

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