A formula for second derivative I want to prove that if $f''$ is continuous at $x_0$, then $$f''(x_0)=\displaystyle\lim_{h\to 0}\dfrac{f(x_0+2h)+f(x_0)-2f(x_0+h)}{h^2}$$
Any hint to prove it?
I can't use l'Hopital or Taylor.
Thanks.

Consider that $f'(x)=\lim\limits_{h\to0} \frac{f(x+h)-f(x)}h$ and $f''(x)=\lim\limits_{h\to0} \frac{f'(x+h)-f'(x)}h$. Subsituting the first into the latter should lead you in the right direction.
