Why Read This Book? - Index of

238 Chapter 7 Functions of a Real Variable
lines) is bounded, then f is uniformly continuous. The converse of Exercise 7.7.3
is not true. Shortly, we will point out a function f that is uniformly continuous on
a set A, but for which inequality (7.39) does not hold across A for any m>0.
Naturally, we want to include a demonstration that a continuous function need
not be uniformly continuous on a set.
EXERCISE 7.7.4 What does it mean for f not to be uniformly continuous on A?
Example 7.7.5 Show that f(x) = 1/x is not uniformly continuous on (0, 1).
Solution First, we do some scratchwork. The vertical asymptote at x = 0 will
provide us with x and y values that are very close together but whose functional
values can differ as much as a strategically chosen ɛ-value. We have to play with
the inequality
� �
�
�
1 1 �
� − �
x y �
= |y − x|
|x||y|
≥ ɛ (7.40)
and find some ɛ>0 so that, regardless of δ>0, we can find two points x and y
that are within δ of each other and satisfy inequality (7.40). No obvious ɛ-value
jumps out at us, so let’s try ɛ = 1 and see if we can proceed.
Inequality (7.40) itself suggests a way to find x and y. Whatever we decide to
let x be, we can let y = x + δ/2 so that |y − x| = δ/2