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I am seeking for a function $[0,1]\to \mathbb{R}$ that is continuous and has derivative almost everywhere, but this derivative is not integrable niether on sense of improper integrals.

For instance, $f(x)=\sqrt{x}$ is s.t. $f'(x)=\dfrac{1}{2\sqrt{x}}$ and $f'$ is Lebesgue integrable. I am seeking an example s.t. $f'(x)$ is not Lebesgue integrable.

This is the difference on the question What is an example that a function is differentiable but derivative is not Riemann integrable

Thank you in advance.

Quiet_waters
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    https://math.stackexchange.com/a/941842/447237 check if this helps –  May 03 '21 at 16:53
  • @Aierel thank you so much. I feel that I have a problem of language here, maybe you could help me. The example of Aubrey in that link you've posted has a finite improper integral https://www.wolframalpha.com/input/?i=integrate_0%5E1+%28+2+x+sin%281%2Fx%5E2%29+-+%282+cos%281%2Fx%5E2%29%29%2Fx%29. I've thinked that this means that $f'(x)$ is Lebesgue integrable, but Aubrey said its not... How can I name the functions that have finite improper integrals...? – Quiet_waters May 03 '21 at 16:58
  • So, I am seeking for functions s.t. derivative is s.t. $1/x$, I mean, the integral diverges... – Quiet_waters May 03 '21 at 16:59

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Define $f$ to be $0$ on the Cantor set $E$, and on each open interval $(a_k,b_k)$ of $[0,1] \backslash E$ take $f(x) = (b_k-a_k)^c (x-a_k)(b_k-x)$ where $-2 < c < -2 + \log_3(2)$. This is differentiable everywhere except on $E$. The condition $c > -2$ ensures that $f$ is continuous, while $c < -2 + \log_3(2)$ ensures that the derivative is not Lebesgue integrable.

Robert Israel
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