5.4 Uniform Continuity 265*corollary 5.4.13 A continuous f: (a,b)-+ ~is uniformly continuous on
(a, b) {::} both lim f(x) and lim f(x) exist.
x--+a+ x-t-b-
*corollary 5.4.14 If f: (a,b)-+ ~is continuous, monotone, and bounded
on (a, b), then it is uniformly continuous on (a, b).
Proof. Theorems 5.2.17 and 5.2.18, and Corollary 5.4.13. •EXERCISE SET 5.4l. Prove directly from Definition 5.4.l that the function f(x) = 2x^2 -5x+2
is uniformly continuous on [-3, l]. Also, on (-2, 2). (See Example 5.4.2.)- Prove directly from Definition 5.4.l that the function f(x) = x^3 is uni-
formly continuous on [O, 3]. Also, on (-2, 1). (See Example 5.4.2.) - Prove Theorem 5.4.3.
- Prove Lemma 5.4.5.
- Prove directly from the c:-o definition that the function f(x) = 7x - 8
is uniformly continuous on R Is this function bounded on~? Does that
contradict Theorem 5.4.6? - Prove directly from the c:-o definition (or its negation) that the function
g(x) = x^2 is not uniformly continuous on R - Prove that the function f(x ) = 1 /x is uniformly continuous on [1,oo).
- Prove that the function f(x ) = 1/x^2 is not uniformly continuous on (0, 1),
first directly from the c:-o definition or its negation, and then as a simple
corollary of a theorem of this section. - Prove that the function f(x) = sinx is uniformly continuous on R [Use
an inequality established in the material leading to Theorem 5.1.16.] - Prove that if f is defined on an interval I and 3M > 0 3 'ilx, y E I ,
lf(x) - f(y)i ::; Mix - yj, then f is uniformly continuous on I. [When
this happens, we say that f satisfies a Lipschitz condition on J.] - Determine which of the following functions f are uniformly continuous
on the given set. [Use the theorems of this section where helpful.]
(a) f(x) = 5x^2 - 3x + 7 on [1, 3]
(c) f(x) = 1/x^2 on (1, 5)
(e) f(x) = tanx on (-~, ~)(b) f(x) = 5x^2 - 3x + 7 on (1, 3)
(d) f(x) = xsinx on (0, ~)
(f) f(x) = tanx on (-i, i)