Let E Rn and suppose that D is dense in E (i.e., that D E

Question:

Let E ⊂ Rn and suppose that D is dense in E (i.e., that D ⊂ E and = E). If f: D → Rm is uniformly continuous on D, prove that f has a continuous extension to E; that is, prove that there is a continuous function g : E → Rm such that g(x) = f(x) for all x ∈ D.