How to prove that a function defined by an inf is differentiable

I have been stuck on this problem for 2 days.

We have a function $f_A$ defined by : $f_A(x)= \text{inf}(|x−a|,a\in A)$ with $A\subset \mathbb{R}$

We fix $x$ consider the special case where there is $\epsilon$ $>0$ such as $]x-\epsilon,x+\epsilon[ \subset A$

We want to show that $f_A$ is differentiable at $x$ and we want the value of $f_A'(x)$

I tried calculating the limit as $h$ approaching $0$ of $(f_A(x-h)-f_A(x)) / h$

According to the statement of your problem, we have that a open interval containing $x$ is contained in $A$. Doesn't that mean there is a neighborhood of $x$ such that $f_A$ is constant with value $0$?