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I was drawing smooth curves through points $(a,f(a))$ and $(b,f(b))$ in an attempt to come up with a proof of the Mean Value Theorem.

As I was doing that, I noticed that I couldn't draw the curve in such a way that its derivative would be discontinous.

I tried to think up a function that is differentiable but whose derivative is discontinous but I got nothing. I did a quick Google search for the same question that I've asked here but all I could find was that differentiability of a function implies continuity of the same which I already know.

If $f(x)$ is differentiable over $(a,b)$, does that mean $f'(x)$ is continuous over $(a,b)$? If not, what are some examples of functions where this is not the case?

Ryan
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  • I am reading through it at the moment. I will let you know once I am done. – Ryan Aug 28 '20 at 01:35
  • I believe the current answer from Brian answers your question. However, it is insteresting that although a derivative need not be continuous, it must satisfy the intermediate value property (Darboux's theorem), which limits the "types" of discontinuities that can occur – whpowell96 Aug 28 '20 at 01:37

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consider if $x\neq 0, $$g(x)=x^2sin(1/x)$ and $g(0)=0.$

Note that the curve you can usually draw is continuously differentiable. But there can be differentiable function but not continuously differentiable.

xyz
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