Difference Equations to Differential Equations (10).pdf

Section
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 Some Consequences
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In this section we consider two properties of functions which are very closely connected
to the notion of continuity. The first of these, the Intermediate Value Theorem, says that
the graph of a continuous function is a connected continuum in the sense of our normal
intuition. That is, the theorem states that as a continuous function changes from one value
to another, it must take on every intermediate value. The second theorem, the Extreme
Value Theorem, says that a continuous function on a closed interval attains a maximum
and a minimum value on that interval. This is related to our intuitive notion that if we
draw a continuous curve with definite beginning and ending points, then the curve has
a point where it is higher than at any other point and a point where it is lower than
at any other point. We shall not attempt formal justifications of these theorems; such
justifications require inquiries into the subtleties of real numbers which are best left to
more advanced courses.
We will begin with a statement of the Intermediate Value Theorem, followed by a
consideration of its application to solving equations.
Intermediate Value Theorem If f is a continuous function on a closed interval [a, b]
and m is any number between f(a) and f(b),