This problem is taken from Problem 2.4.31 (page 84) from Problems in Mathematical Analysis: Integration by W. J. Kaczor, Wiesława J. Kaczor and Maria T. Nowak.
Give an example of a bounded function $f:[0,1] \to \mathbb{R}$ which is not Riemann Integrable, but is a derivative of some function $g$ on $[0,1]$.