Proof identity of differential equation I would appreciate if somebody could help me with the following problem:
Q: $f''(x)$ continuous in $\mathbb{R}$ show that 
$$ \lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=f''(x)$$ 

Hint: what is the limit definition of a derivative? Try writing it out, first applying it to $f(x)$, and then applying it to $f'(x)$. Can you apply this definition several times to get what you are looking for? (Note that you must be careful in this argument, as it requires the continuity of $f''(x)$!)
