Exercise 4 4 2 A Is F X 1 X Uniformly Continuous On 0 1 B Is G X X 2 1 Uniformly

Exercise 4.4.2. (a) Is f(x) = 1/x uniformly continuous on (0, 1)?

(b) Is g(x) = √ (x ^2 + 1) uniformly continuous on (0, 1)?

(c) Is h(x) = x sin(1/x) uniformly continuous on (0, 1)?

Exercise 4.4.3. Show that f(x) = 1/x 2 is uniformly continuous on the set [1,∞) but not on the set (0, 1].

Exercise 4.4.4. Decide whether each of the following statements is true or false, justifying each conclusion.

(a) If f is continuous on [a, b] with f(x) > 0 for all a≤x≤b, then 1/f is bounded on [a, b] (meaning 1/f has bounded range).

(b) If f is uniformly continuous on a bounded set A, then f(A) is bounded.

(c) If f is defined on R and f(K) is compact whenever K is compact, then f is continuous on R. Exercise

4.4.5. Assume that g is defined on an open interval (a, c) and it is known to be uniformly continuous on (a, b] and [b, c), where abc. Prove that g is uniformly continuous on (a, c).

Exercise 4.4.6. Give an example of each of the following, or state that such a request is impossible. For any that are impossible, supply a short explanation for why this is the case.

(a) A continuous function f : (0, 1)→R and a Cauchy sequence (x n) such that f(x n) is not a Cauchy sequence;

(b) A uniformly continuous function f : (0, 1)→R and a Cauchy sequence (x n) such that f(x n) is not a Cauchy sequence;

(c) A continuous function f : [0,∞)→R and a Cauchy sequence (x n) such that f(x n) is not a Cauchy sequence;

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