Let $f$ be a real function defined on $\mathbb{R}$ satisfying the condition:
$$\lim_{h\rightarrow 0} [f(x+h)-f(x-h)]=0 \mbox{}$$
for any $x\in\mathbb{R}$. Does it imply continuity of $f$?
I believe the counterexample to that statement is the following function:
$$g(x)=\mathbb{1}_{\{0\}}\mbox{,}$$ which is clearly not continuous at $x=0$.
