Introduction to Mathematical Analysis I - 3rd Edition

88 3.5 Uniform Continuity Exercises 3.5.1 Prove that each of the following functions is uniformly continuous on the given domain: (a) f (x) =ax+b, a,b∈R, onR. (b) f (x) =1/x on [a,∞), where a>0. 3.5.2 ▶Prove that each of the following functions is not uniformly continuous on the given domain: (a) f (x) =x2 on R. (b) f (x) =sin 1 x on (0,1). (c) f (x) =ln(x) on (0,∞). 3.5.3 Determine which of the following functions are uniformly continuous on the given domains. (a) f (x) =xsin(1 x) on (0,1). (b) f (x) = x x+1 on [0,∞). (c) f (x) = 1 |x−1| on (0,1). (d) f (x) = 1 |x−2| on (0,1). 3.5.4 Let D⊂Randk ∈R. Prove that if f ,g: D→Rare uniformly continuous onD, then f +g andk f are uniformly continuous on D. 3.5.5 Give an example of a subset Dof Rand uniformly continuous functions f,g: D→Rsuch that f gis not uniformly conitnuous onD. 3.5.6 Let Dbe a nonempty subset of Rand let f : D→R. Suppose that f is uniformly continuous on D. Prove that if {xn}is a Cauchy sequence with xn ∈Dfor every n∈N, then {f (xn)}is also a Cauchy sequence. 3.5.7 ▷Let a,b∈Rand let f : (a,b) →R. (a) Prove that if f is uniformly continuous, then f is bounded. (b) Prove that if f is continuous, bounded, and monotone, then it is uniformly continuous. 3.5.8 ▷Let f be a continuous function on[a,∞). Suppose lim x→∞ f (x) =c. (a) Prove that f is bounded on [a,∞). (b) Prove that f is uniformly continuous on[a,∞). (c) Suppose further that c >f (a). Prove that there exists x0 ∈[a,∞) such that f (x0) =inf{f (x) : x ∈[a,∞)}.

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