# Does there exist such an interval?

If $f: \mathbb{[a,b]} \to \mathbb{R}$ and $f$ is twice differentiable at a point $c$. Does there exist an interval $[p,q]$ in $[a,b]$ where $f$ is differentiable

I think here, $f'$ is continuous at $c$.Then by the definition of continuity there must exist such an interval

• How do you define "twice differentiable at $c$" if $f$ is not differentiable on a neighborhood of $c$? – Clement C. Aug 10 '18 at 19:13

However, since $f$ is twice differentiable at $c$, that means that its derivative at point $c$ is differentiable.
Now assume that there are no intervals around $c$ that are differentiable. Then, the derivative of $f$ could not be shown to be differentiable, since $f'$ wouldn't even be continuous at $c$. Hence, the statement has been shown.