Generalized Uniform Continuity?

I know that if $X$ is compact metric space then a continuous map $F:X\rightarrow Y$ is uniformly continuous and if $X$ not compact then it is not always true. But I was wonder if something for more general holds like:

If $X$ and $Y$ are metric spaces and $F:X\rightarrow Y$ is continuous and $C\subseteq X$ is closed, does it follow that for every continuous $\epsilon:X\rightarrow (0,\infty)$, there exists a continuous $\delta:C\rightarrow (0,\infty)$ sucht that if $c\in C$ and $d(x,c)<\delta(c)$ then $d(F(x),F(c))<\epsilon(c)$?