C ONTINUITY (PA RT III) THE INTE R M E DIATE VALU E THEO R E M In this tutorial, we take a look at a very important property of every typical continuous function. This theorem is called the Intermediate Value Theorem. Did you know that the working principle of a graphing calculator is based on the Intermediate Value Theorem? Well, it is , and later on, we'll briefly see how a typical graphing calculator makes use of this Theorem. You can read more about the Theorem at Wikipedia. You can also view Java Applets illustrating the Theorem at calculusapplets.com. This Theorem is relatively straightforward, and since there isn't any easier way to define it, here's a definition taken directly from the textbook:
Suppose f is continuous on the closed interval [a, b] and let N be any number between f(a) and f(b). Then there exists a number c in (a, b) such that f(c) = N.
Note that this Theorem applies to continuous functions only. The graph below illustrates the Theorem:
A GEOMETRIC INTERPR ETATION OF THE INTERMEDIATE VALUE THEOREM If you read the Intermediate Value Theorem carefully, you'll find that it is quite easy to believe. Suppose we have a horizontal line y = N
between f(a) and f(b) [on the next page]. Based on the principle of the
Intermediate Value Theorem, we say that, since f is continuous, then the line y = N WILL DEFINITELY touch f somewhere, and that point of intersection will be (c, f(c)), or (c, N).