Suppose that E ⊂ X and that f : E → Y.
(a) If f is uniformly continuous on E and xn ∈ E is Cauchy in X, prove that f(xn) is Cauchy in Y.
(b) Suppose that D is a dense subspace of X (i.e., that D ⊂ X and  = X). If Y is complete and f: D → Y is uniformly continuous on D, prove that f has a continuous extension to X; that is, prove that there is a continuous function g: X → Y such that g(x) = f(x) for all x ∈ D.