266 Chapter 5 • Continuous Functions
- Prove that the function f(x) = .jX is uniformly continuous on [O, + oo).
[Hint: first prove that lvx -v'YI ::::: JIX=Yi.J - Prove that the sum of two functions that are uniformly continuous on a
set A is uniformly continuous on A. - Prove by counterexample that the product of two functions uniformly
continuous on a set A need not be uniformly continuous on A. - Use the theorems of this and previous sections to prove that the functions
tan x and sec x are continuous, but not uniformly, on ( -~, ~), while cot x
and cscx are continuous, but not uniformly, on (0,7r).
16. Find an example of an interval (a, b) and a function f : (a, b) __, JR that
is continuous and bounded, but not uniformly continuous.- Prove that if f: [a,oo) __,JR is continuous and lim f(x) is finite, then f
X->00
is uniformly continuous on [a, oo).
1 8. Suppose f and g are uniformly continuous on a set A. Prove that
(a) if f and g are bounded on A, then f g is uniformly continuous on A;
(b) if A is bounded, then f g is uniformly continuous on A.- Prove that if f : V(f) --> JR is uniformly continuous on A, and g is
uniformly continuous on J(A), then go f is uniformly continuous on A. - Show by counterexamples that Theorem 5.4.10 is not true if "uniformly
continuous" is replaced by "continuous,'' or if A is not a bounded set. - Prove that the converse of Exercise 10 is not true, by showing that the
function f(x) = .jX is uniformly continuous on [O, l] but does not satisfy a
Lipschitz condition there. [Thus, a Lipschitz condition is strictly stronger
than uniform continuity.] - Suppose f : JR __, JR is periodic with period p > 0. That is, \:/x E JR,
f ( x + p) = f ( x). Prove that if f is continuous on any compact interval of
the form [a, a + p], it must be bounded and uniformly continuous on R - Suppose a < b < c < d. Prove that if f is uniformly continuous on (a, b)
and on (c , d) then f is uniformly continuous on (a, b) U (c, d). Prove that
the same is true if the intervals are closed, even when b = c. - Based on the result of Exercise 22, one might make the following con-
jecture: If f is uniformly continuous on disjoint sets A and B, then f is
uniformly continuous on A U B.
(a) Find a function f and two bounded open intervals that prove this
conjecture false.