Let $a<b$ and $f:(a,b)\to \mathbb R$ be continuous and such that for all $x\in (a,b)$, $\lim_{h\to 0} \frac{f(x+h) - 2f(x) +f(x-h)}{h^2}$ exists. Let $\beta:x\mapsto \lim_{h\to 0} \frac{f(x+h) - 2f(x) +f(x-h)}{h^2}$. If $\beta$ is continuous over $(a,b)$, prove that $f\in C^2((a,b))$ (i.e. $f$ is twice continuously differentiable)
I'm aware that if $f$ is twice differentiable, then $\beta = f''$. I'm also aware that the mere existence of $\beta$ does not guarantee that $f$ is twice differentiable. The strong assumption in this problem is that $\beta$ be continuous. The continuity assumption on $f$ should also come into play somewhere...
I've seen this problem asked at an oral exam and I find it quite interesting. I've been thinking about it for a few days but I have made zero progress towards a solution ...
Hint: Use a Rolle's theorem style argument to show that if $\beta$ is a positive constant then $f$ cannot have a local maximum in $(a,b).$ Use this to show that $f$ is affine if $\beta$ is identically zero. Use this to show that $f\in C^2$ for general $\beta.$