Explaining the Intermediate Value Theorem

Date: 02/18/2004 at 20:21:08
From: Laura
Subject: the Intermediate Value Theorem

Why is the Intermediate Value Theorem true?  I understand it, but how
can I EXPLAIN why it is true?  Just providing examples isn't 
sufficient, and I am only in a basic calculus class so an advanced
proof won't help me.  I just want to know how to explain the Theorem
in its most simple form:

If f is a continuous function on the closed interval [a,b] and N is
any number between f(a) and f(b), there must be a number c in (a,b)
such that f(c)= N.

Date: 02/18/2004 at 21:00:30
From: Doctor Rob
Subject: Re: the Intermediate Value Theorem

Thanks for writing to Ask Dr. Math, Laura!

The explanation is that the graph of a function continuous on an
interval is connected.  You can draw it without lifting your pencil.
That means that there are no jumps or gaps, so as x varies from a to
b, and f(x) varies from f(a) to f(b), f(x) cannot jump over or miss
the value of N.  The continuity of the function guarantees that it
must contain every value between f(a) and f(b), including N.

Feel free to reply if I can help further with this question.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/