I'm kinda stuck on this problem. Would someone give me a hand?

Suppose $\displaystyle f:[a,b] \rightarrow R$ is differentiable and $\displaystyle f'$ is differentiable at $\displaystyle t \in (a,b)$. Show that

$\displaystyle f''(t)=\lim_{h\to 0} \frac{f(t+h)-2f(t)+f(t-h)}{h^2}$

Is there a function that satisfies this limit but $\displaystyle f'$ is not differentiable at $\displaystyle t$?