Let $X$ be a locally compact Hausdorff space and $K$ be a compact subset of $X$. Then any function $f\in C(K)$ can be extended to a function in $C(X)$ which vanishes outside a compact set.
I have searched several books for the proof. It seems that the authors believe it is just an exercise-level proposition. Since $X$ is an LCH space, we can always find open set $V$ containing $K$ with compact closure $\overline{V}$ and we can extend $f$ to $\overline{V}$ by classical Tietze extension theorem. I also know the fact that $f\in C(K)$ implies that the the range of $f$ is contained in a closed interval $[a,b]$, but how do we define a $F \in C(X)$ satisfying the requirements?
Best Answer
Extend $f$ to $\hat f:\operatorname{cl}V\to\Bbb R$ just as you’ve already done. If $V$ is clopen in $X$, let
$$f^*:X\to\Bbb R:x\mapsto\begin{cases} \hat f(x),&\text{if }x\in V\\ 0,&\text{otherwise}\;. \end{cases}$$
Otherwise, apply Uryson’s lemma to the normal space $\operatorname{cl}V$ to get a continuous $g:\operatorname{cl}V\to[0,1]$ such that $g(x)=1$ for all $x\in K$, and $g(x)=0$ for all $x\in(\operatorname{cl}V)\setminus V$. Then define
$$f^*:X\to\Bbb R:x\mapsto\begin{cases} g(x)\hat f(x),&\text{if }x\in\operatorname{cl}V\\ 0,&\text{otherwise}\;. \end{cases}$$