Let $f:(a.b)\rightarrow \mathbb R$ be a continuous function. How to prove that if for $\varepsilon >0$ there is a $\delta >0$ such that for $x\in (a,b)$, $|h|< \delta$ such that $x+2h \in (a,b)$
$$\left|\frac{f(x+2h)-2f(x+h)+f(x)}{h^2}\right|<\epsilon, $$
then $f$ is of the form $f(x)=\alpha x+\beta$, where $\alpha,\beta$ are constants ?