2
$\begingroup$

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

$\endgroup$
  • 3
    $\begingroup$ 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
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.