An extension of a continuous function onto the closure

Question

Let $S$ be a proper subset (not closed) of a metric space $X$. Suppose that $f:S \to \Bbb R$ be continuous. I want to know the condition under which $f$ can be continuously extended to $\bar{S}$.

I am only aware of one condition when $f$ is uniformly continuous. Is there any other ?

Can't this always be done? Since a set is dense in its own closure? — GPerez, Jan 09 '15 at 16:16
@GPerez: So! let $x \in \bar{S}-S$. Then there is a sequence ${x_n} \in S$ which converges to $x$. Then what should we define as $f(x)$. There is no guarantee that ${f(x_n}$ converge as $x \not \in S$. — Anonymous, Jan 09 '15 at 16:39
Hmm, I'll think about it some more then. It does work with the additional assumption that $X$ is complete though. — GPerez, Jan 09 '15 at 16:43
@GPerez: Even that is not enough. To make sure that ${f(x_n)}$ cauchy when ${x_n}$ cauchy we need uniform continuity of $f$. — Anonymous, Jan 09 '15 at 16:48
Sorry, I was getting confused with this proof, of a similar situation, but not really that similar I guess. — GPerez, Jan 09 '15 at 18:02
Note that if $X$ is compact, then such a function extends if and only if it is uniformly continuous. — Prahlad Vaidyanathan, Apr 30 '15 at 06:13

score 7 · Answer 1 · answered Apr 30 '15 at 06:11

This question was investigated in my joint with Misha Mitrofanov paper “Approximation of continuous functions on Fréchet spaces” (at pages 794-796 of English version, which is downloadable in the source file). In particular, there I proved the following

Lemma 2. Let $X$ be a Fréchet-Urysohn topological space, $Y$ be a regular topological space, and $D$ be a dense subset of the space $X$. A continuous mapping $f : D\to Y$ can be extended to a continuous mapping $\hat f : X\to Y$ iff for each sequence $\{x_n\}$ of points of the set $D$ convergent in $X$, a sequence $\{ f(x_n)\}$ is also convergent.

which holds when both spaces $X$ and $Y$ are metric, because metric spaces are Fréchet-Urysohn and regular.

An extension of a continuous function onto the closure

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