Non integrable function

Question

There are some functions whose indefinite integration is not possible. What is the reason behind this?

Answer 1

There are several possible reasons. For instance, Darboux's theorem says that if $f$ is differentiable, then $f'$ has the mean value property. Therefore, a function that lacks that property cannot have an indefinite integral. That's the case, for instance, of the function$$f(x)=\begin{cases}1&\text{ if }x\geqslant 0\\0&\text{ otherwise.}\end{cases}$$