Is there $g:X\to (0, \infty)$ s.t $g(x)\leq \inf\{f(y): d(y, x)<f(x)\}, \forall x\in X$

Question

Let $(X, d)$ be a metric space and $f:X\to (0, \infty)$ be a continuous function.

Is there a continuous function $g:X\to (0, \infty)$ such that

\begin{equation} g(x)\leq \inf\{f(y): d(y, x)<g(x)\}, \forall x\in X \end{equation}

Answer 1

Not necessarily. Let $X=((0,\infty),|\cdot|)$, and let $f(x)=x$, then for all $x\in(0,\infty)$ we have that $\inf\{f(y):d(y,x)<f(x)\}=\inf\{y:|y-x|<x\}=0$, hence $g\leq0$, but $g:X\rightarrow(0,\infty)$.

If you allow $g:X\rightarrow[0,\infty)$, then clearly $g(x)=0$ satisfies the conditions.