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, on R. (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. 1 (b) f (x)= sin on (0,1). x (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 ) on (0,1). x x (b) f (x)= on [0,∞). x + 1 1 (c) f (x)= on (0,1). |x − 1| 1 (d) f (x)= on (0,1). |x − 2| 3.5.4 Let D ⊂ R and k ∈ R. Prove that if f,g: D → R are uniformly continuous on D, then f + g and kf are uniformly continuous on D. 3.5.5 Give an example of a subset D of R and uniformly continuous functions f,g: D → R such that fg is not uniformly conitnuous on D. 3.5.6 Let D be a nonempty subset of R and let f : D → R. Suppose that f is uniformly continuous on D. Prove that if {xn} is a Cauchy sequence with xn ∈ D for every n ∈ N, then { f (xn)} is also a Cauchy sequence. 3.5.7 ▷ Let a,b ∈ R and 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 f (x)= c. x→∞ (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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