I have some elementary problems with understanding the concept of "integrability".
The lecture notes I'm reading at one points state that some function like $f_{\operatorname{step}}(x) = \lfloor 4x \rfloor$ is "Riemann integrable" on $[0, 1]$ because it is a step function. At another point they state "Darboux's theorem": if $f [a,b] \rightarrow \mathbb{R}$ is differentiable, then $f'$ will assume every value between $f'(a)$ and $f'(b)$.
Isn't this a contradiction? If $f_{\operatorname{step}}$ is integrable, it must be some function's derivative and as such should take on every value between $f_{\operatorname{step}}(0)$ and $f_{\operatorname{step}}(1)$ which it clearly doesn't (i.e. any non-integer value). Is Riemann integrability not the same as the "ordinary" integrability I know from Calculus? (There are similar questions on this site but after reading them I still don't quite get it).