1549901369-Elements_of_Real_Analysis__Denlinger_

5.4 Uniform Continuity 257

We now consider a little more closely what it means for a function to be contin-

uous on a set. We shall refine this concept and define a slightly stronger form

of continuity on a set, called "uniform" continuity. This will have special sig-

nificance in Chapter 7, when we study the Riemann integral. First recall what

it means for a function to be continuous on a set.

Recall: A function f : D(f) ---t IR is continuous on a set A <;;;; D(f) if it is

continuous at each point of A; that is

Vx EA, Ve> 0, :Jo> 0 3 Vy E D(f), Ix - YI < O => lf(x) - f(y)I < e.

We are now ready to define the stronger form of continuity. You will have

to look very carefully to see the difference.

Definition 5.4.1 A function f : D(f) ___, IR is uniformly continuous on a

set A <;;;; D(f) if

Ve> 0, :Jo> 0 3 Vx,y EA, Ix -yl < o => lf(x)-f(y)I < e.

The difference between these two types of continuity on a set A is quite

subtle. It lies in the order of quantification. In the definition of "continuous on

A," for every choice of x and e there exists a o = o(e, x), dependent on both

e and x. In the definition of "uniform continuity" on A, for every choice of e

there exists a o = o(e) that works for all x in A (independent of the choice of

x, and dependent only one).

Although the difference is subtle, it is very significant. It is perhaps easier

to see the difference if we outline the differing strategies we would use to prove

each:

To prove that f is continuous on A: Let x E A, and let e > 0.

Find O > 0 3 Vy E D(f), Ix - YI < O => lf(x) - f(y)I < e.

To prove that f is uniformly continuous on A: Let e > 0.

Find o > 0 3 Vx, y EA, Ix - YI < O => lf(x) - f(y)I < e.

To prove that f is continuous on A, for each x E A and each e > 0 we

must find a o > 0, depending on x and e, that makes a certain implication

true for ally E D(f). To prove that f is uniformly continuous on A , for each

e > 0 we must find a o > 0, depending only on e, that makes the implication

true for all x, y E A. Thus, it seems that uniform continuity is stronger than

continuity. Indeed, Theorem 5.4.3 below says that for f : A ___, IR, uniform

continuity implies continuity.

Example 5.4.2 Prove that the function f(x )
.continuous on the interval [-1, 5].

3x^2 - 2x - 1 is uniformly