Complex Variable

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Image text transcribed for accessibility: Suppose f: X rightarrow Ohm is uniformly continuous; show that if {xn} is a Cauchy sequence in X then {f (xn)} is a Cauchy sequence in ohm. Is this still true if we only assume that f is continuous? (Prove or give a counterexample.) Recall the definition of a dense set (1.14). Suppose that ohm is a complete metric space and that f: (D, d) rightarrow (ohm; rho) is uniformly continuous, where D is dense in (X, d). Use Exercise 5 to show that there is a uniformly continuous function g: X rightarrow ohm with g(x) = f (x) for every x in D.

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  • TrueGrit
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