# Real analysis

1. Nov 17, 2013

### Lee33

1. The problem statement, all variables and given/known data

Let $S\subset E$ where $E$ is a metric space with the property that each point of $S^c$ is a cluster point of $S.$ Let $E'$ be a complete metric space and $f: S\to E'$ a uniformly continuous function. Prove that $f$ can be extended to a continuous function from $E$ into $E'$ and that this extended function is also uniformly continuous.

2. The attempt at a solution

This is a complicated problem for me and I want to know if my reasoning is correct?

Let $s\in S^c$ then since $s$ is a cluster point of $S$ there exists a sequence $s_n \in S$ such that $s_n \to s$ thus $s_n$ is a Cauchy sequence. Since $f$ is a uniformly continuous function and $E'$ is complete then $f(s_n)$ is also a Cauchy sequence that converges in $E'$ to some $x,$ thus we have $\lim f(s_n) = x.$

Now we want to show that $f$ can be extended to $E$ and that this extension is uniformly continuous, in order to do that we must have that for any sequence in $E$ that converges to $s$ then the image sequence must converge to the same limit $x.$ Therefore we define our extension to be $\lim g(y_n) = x$ for $y_n \in E.$

Now to prove that this is indeed an extension we must show two things: that is uniformly continuous and that the image sequence converges to the same limit $x$ thus $\lim g(y_n) = x.$

Is my reasoning correct? Thank you for your time!