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This is motivated by a question I saw elsewhere that asks whether there is a real-valued function on an interval that contains no monotone subintervals.

Edit: Note that I am asking for a function whose derivative exists but is not continuous anywhere.

Martin Sleziak
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Zachary F
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2 Answers2

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(Adding back someone else's answer that was deleted for some reason. )

Answer: NO: If $f$ is differentiable everywhere on $\mathbb R$, then $f'$ is continuous somewhere.

Suppose $f$ is differentiable everywhere. Then $f$ is continuous everywhere. The functions $$ g_n(x) = \frac{f\left(x+\frac{1}{n}\right)-f(x)}{\frac{1}{n}} $$ are continuous and converge pointwise everywhere to $f'(x)$. Therefore $f'(x)$ is of Baire class $1$, and therefore has lots of points of continuity.

GEdgar
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Consider the Weierstrass Function

It is continuous everywhere and only differentiable at a set of points with measure 0. I don't know if that suffices for you, but I think it is quite amazing already.

RGS
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    I thought of that too at first, but I think the question assumes implicitly that the function is differentiable. –  Nov 16 '16 at 00:46
  • Do you have a reference for your statement of the Weierstrass function? What is the characterization of the set? – user48672 May 18 '17 at 22:34
  • user48672 if you follow the link in the answer, you will see my reference – RGS May 19 '17 at 08:09