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.
Best Answer
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.