5.3 Continuity on Compact Sets and Intervals 247That is , the function !IA is the same as the function f, except that its domain
has been "restricted to" A. Thus the expressions f : A --> IR and flA : A --> IR
mean the same thing.
With this subtlety behind us, we are ready to discuss continuity of functions
on compact sets and intervals.CONTINUITY ON COMPACT SETS
(EXTREME VALUE THEOREM)Compact set s were introduced and discussed more fully in Section 3.3. For
our purposes, and for anyone who omitted Section 3.3, the following definition
is sufficient.Definition 5.3.3 A set A <;;;: IR is said to be a compact set if it is closed and
bounded.The following are examples of compact sets:
(a) finite sets;
(b) closed intervals of the form [a, b];
(c) U : n EN} U {O};
(d) {xn: n EN} U {L}, where Xn--> L ;
( e) unions of finitely many of the above.Theorem 5.3.4 Every nonempty compact set has a maximum and a mini-
mum.
Proof. Exercise 3. •Theorem 5.3.5 (Sequential Criterion for Compactness)^10 A set A of
real numbers is compact if and only if every sequence of points of A has a
subsequence that converges to a point of A.
Proof. Let A be a set of real numbers.
Part 1 (::::}): Suppose A is compact. Let {an} be a sequence of points of
A. Now A is a bounded set, so {an} is a bounded sequence. By the Bolzano-
Weierstrass Theorem for sequences, {an} has a convergent subsequence { ank}.
Let L = lim ank. Now A is closed, since it is compact. So, by the sequential cri-
k-+oo
terion for closed sets (3.2.19), LE A. Thus, {an} has a subsequence converging
to a point of A.
- For readers who did not study Section 3 .3, this theorem and its proof are repeated ver-
batim from 3.3.13.