For every closed set $F$ in a completely regular space $X,$ does there exist a nonzero continuous function $f:X\to [0,1]$ such that $f=0$ outside $F?$

Question

Let $X$ be a completely regular space, that is, for every closed set $F\subseteq X$ and $x\not\in F,$ there exists a continuous function $g:X\to [0,1]$ such that $g(F) = \{0\}$ and $g(x) =1.$

Question: For every closed set $F\subseteq X,$ does there exist a nonzero continuous function $f:X\to [0,1]$ such that $f=0$ outside $F?$

Answer 1

Not true even in the real line. If $F$ is a singleton set then any continuous function vanishing outside $\{x\}$ is identically $0$.