Let be a subset of , and be uniformly continuous, where is complete. Show that there exists a continuous function which maps the closure of to such that
So basically, we are trying to prove that a function can be extended to its domains closure, and still retain the same values on the original set.
It's late, so maybe I need to spend more time pondering this question, but I'm frankly a little stumped - of the continuity problems I have tackled so far from chapter 4 of Rudin, this is the first involving the hypothesis of a mapping into a complete metric space, so it suggests (at first to me anyway) a different method of proof.
I guess I would start off with first establishing that the closure of E is (obviously) a closed set, with a limit point in X (in fact, in E closure by definition). I can than approach this limit point with a convergent Cauchy sequence, and then the image of this sequence will again be Cauchy in Y and thus convergent in Y. These facts follow from the hypothesis that the function f is uniformly continuous, and Y is complete. Then I suppose I could somehow use the sequential characterization of continuity to conclude the theorem...but rigorizing this argument, and then actually proving that this "extension" big F is equal to little f on E, is something I'm not sure about.
Anyway, any other ideas, or modifications to mine, for this problem?