1549901369-Elements_of_Real_Analysis__Denlinger_

5.4 Uniform Continuity 261

For each i, /xi-Xi-1/ = Xi-Xi-1 = [a+ ib~a] - [a+ (i - l)b~a] = b~a <

o. To see that f is bounded on [xi-l, xi], for each i, first select a point xi E V(f)

in [xi-1, xi] if such a point exists. Thus, when [xi-l, xi] n V(f) -j. 0,

y E [xi-1, xi] n V(f) => /y - xi I :::; /xi -Xi-1 /

:::? /y-xi/< o

n

* /f(y) - f(xi)/ < 1 by (8)

=> f(xi) - 1 < f(y) < f(xi) + 1

* /f(y)/ < /f(xi)/ + 1.

Now, A~ [a,b] = LJ[xi-1,xi]· Thus, Vy EA, /f(y)/ < max{/f(xi)I + 1:

i=l

[xi-1, xi] n V(f) # 0}. Therefore, f is bounded on A. •

1
Second proof that the function f(x) = - is not uniformly contin-
x
uous on (0, 1):
This function is unbounded on (0, 1). Hence, the conclusion follows easily
from Theorem 5.4.6. •

The next result shows another special property of compact sets. It is often
very useful, as in proving the Riemann integrability of continuous functions, in
Chapter 7.

Theorem 5.4. 7 If f : A ----; JR is continuous on a compact set A, then f is
uniformly continuous on A.

Proof. Suppose A is compact and f : A ----; JR is continuous. For contra-
diction, suppose f is not uniformly continuous on A. By Lemma 5.4.5, this
means

:Jc:> 0 3 Vo> 0,3x,y EA 3 lx-yl < o but lf(x)-f(y)I ~ c:.

Keep this c: > 0 fixed. Then, Vn EN,

By the squeeze principle, Xn - Yn ----; 0. Also, the sequence { Xn} is "in" the
compact set A, so by the Balzano-Weierstrass Theorem, it has a convergent
subsequence, { Xnk }; say Xnk ----; L.
Now, Vk EN, Ynk = (Ynk - Xnk) + Xnk ----; 0 + L. Hence, Ynk ----; L.