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

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

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.

  • $\begingroup$ Which is so obvious now that you've said it, that I dk why I didn't think of it. $\endgroup$ – DanielWainfleet Aug 10 '18 at 19:20
  • $\begingroup$ Basically, the answer is "yes, by definition of twice differentiable." $\endgroup$ – Clement C. Aug 10 '18 at 19:21
  • $\begingroup$ @ClementC. Yea essentially lol $\endgroup$ – Don Thousand Aug 10 '18 at 19:21

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