I am working on a problem finding an error of an example that uses integration by parts.
The condition of the example says that $f'$ and $g'$ exist almost everywhere.
If $f'$ and $g'$ exist almost everywhere, does this condition implies that $f'$ and $g'$ are Riemann integrable? What I want to argue is that they are not, since $f'$ and $g'$ may not be continuous a.e..
Also, I recall that $f$, $g$ must be differentiable (I suspect this condition is stronger than $f'$ and $g'$ exist a.e.) in order to use integration by part. Can this be a reason for why integration by part is not working on the example that I am working on?