If $\lim_{h \to 0}[f(x+h)+f(x-h)-2f(x)] = 0$ for every $x \in \Bbb R$, does it follow that $f$ is continuous?
I start by rearranging it to be $\lim_{h \to 0}f(x+h)+f(x-h)=\lim_{h \to 0}2f(x)$ and I feel as though I need to rearrange the LHS, but am unsure as to how to proceed.
Thank you in advance
Counterexample: $$ f(x) = \begin{cases} 0 & x < 0\\ 1 & x = 0\\ 2 & x > 0 \end{cases} $$