If the derivative $f'$ exists everywhere then it is shown here that there exist intervals on which $f'$ is Lebesgue integrable. But perhaps there is a function $f$ such that $f'$ only exists almost everywhere and $f'$ is not Lebesgue integrable on any interval?
Basically I'm trying to make sense of a remark on p. 111 of Folland's Real Analysis: "it is not hard to construct examples in which the singularities of $f'$ are so complicated that $f'$ is not Lebesgue integrable on any interval. In this situation the Lebesgue integral is simply insufficient. However, the Henstock-Kurzweil integral [...] is powerful enough to integrate such $f'$ [...]."
