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
The answer is yes, but not for the reason you've provided. Continuity is not as strong of a claim as differentiability is, as the latter implies the former but not the other way around.
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.