Yet Another Calculus Text, 2007a

22 CHAPTER 1. DERIVATIVES
1.5.4 Consequences of continuity
Continuous functions have two important properties that will play key roles
in our discussions in the rest of the text: the extreme-value property and the
intermediate-value property. Both of these properties rely on technical aspects
of the real numbers which lie beyond the scope of this text, and so we will not
attempt justifications.
The extreme-value property states that a continuous function on a closed
interval [a, b] attains both a maximum and minimum value.
Theorem 1.5.11. If f is continuous on a closed interval [a, b], then there exists
arealnumberc in [a, b] for which f(c) ≤ f(x) for all x in [a, b] andarealnumber
d in [a, b] for which f(d) ≥ f(x) for all x in [a, b].
The following examples show the necessity of the two conditions of the theorem
(that is, the function must be continuous and the interval must be closed
in order to ensure the conclusion).
Example 1.5.8. The function f(x) =x 2 attains neither a maximum nor a
minimum value on the interval (0, 1). Indeed, given any point a in (0, 1), f(x) >
f(a) whenever a