Question

A subset D of a metric space (X, d) is dense if every member of X is a limit of a sequence of elements from D.

Suppose (X,d) and (Y,ρ) are metric spaces and D is a dense subset of X.

1. Prove that if f : D -» Y is uniformly continuous then there exists an extension15 of f to a if dn (E D) e X define 7(x) lim f(d,) uniformly continuous function f:X * Y. Hint: 2. Show the result is not true if "uniformly continuous" is everywhere replaced by "continuous".

Answer 1

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