Let $\displaystyle I\subset \mathbb{R}$ be an open interval, let $\displaystyle f:I \rightarrow \mathbb{R}$ be differentiable on $\displaystyle I$, and suppose $\displaystyle f''(a)$ exists at $\displaystyle a\in I$. Show that

$\displaystyle f''(a) = \lim_{h\to 0}\frac{f(a+h)-2f(a)+f(a-h)}{h^2}$

Give an example where this limit exists, but the function does not have a second derivative at $\displaystyle a$ .