KöMaL 2018 A723

Let $f\colon \mathbb{R}\to \mathbb{R}$ be a continuous function such that the limit $g(x)=\lim_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}$ exists for all real $x$. Prove that $g(x)$ is constant if and only if $f(x)$ is a polynomial function whose degree is at most $2$.