If $f:\mathbb{R}\rightarrow\mathbb{R}$ is a differentiable funcion and A is a closed interval, it is not true that $f'$ is Riemann integrable in A. The question is:
•Is there a definiton of integral that makes $f'$ integrable in any case? (Maybe Lebesgue?)
•Could you provide some examples in wich integrability of $f'$ fails for different definitions of integral?
Thank you!
For example, the function $$f(x) = \cases{0 & if $x=0$\cr x^2\cos(1/x^2) & if $x \ne 0$\cr}$$
is differentiable everywhere, but $f'$ is not Lebesgue integrable in a neighbourhood of $0$.