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/
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