Here is a proof that I am having trouble getting started with. I'd appreciate any help.

Let $\displaystyle D$ be a subset of $\displaystyle \mathbb{R}$. Let $\displaystyle f

\rightarrow \mathbb{R}$ be uniformly continuous. Let $\displaystyle x_0$ be a limit point of $\displaystyle D$. Suppose $\displaystyle x_{0}\notin D$. Prove there is a continuous function $\displaystyle g

\bigcup \{x_0\}\rightarrow \mathbb{R}$, such that $\displaystyle g(x)=f(x)$ for all $\displaystyle x\in D$.

I'll be happy to attempt the proof. If anyone could maybe just tell me exactly what direction I should be looking to go I would would appreciate it.