I'm doing some self-study and I ran into a situation as follows. Suppose $(X,d)$ is a metric space and $F\subset X$ is compact. For some $\varepsilon>0$ let $V=\{x:d(x,F)<\varepsilon\}$. Is the function $f:X\to[0,1]$ \begin{align*} f(x)=\frac{d(x,V^{c})}{d(x,V^{c})+d(x,F)} \end{align*} Compactly supported? Or atleast uniformly continuous?
Thanks in advance.