Let $S$ be a subset of the metric space $E$ 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 $E'$ in one and only one way, and that this extended function is also uniformly continuous.
Let $s \in S^c$ then since $s$ is a cluster point of $S$ then there is a sequence $s_n \in S$ that converges to $s$ thus the sequence $s_n$ is a Cauchy sequence. Since $f$ is uniformly continuous and $E'$ is complete we have that $f(x_n)$ is also a Cauchy sequence which converges in $E'$ to some $x.$
How can I now porve that $f$ can be extended, in one and only one way, to a continuous function from $E$?