Bounded derivatives which are not Riemann integrable

In an elementary calculus course, we talk mostly, or exclusively, about integrating continuous, real-valued functions. Since continuous functions on closed intervals are integrable, the Fundamental Theorem of Calculus gives us a method to calculate these integrals (given that we can find an antiderivative). Furthermore, the Fundamental Theorem of Calculus states that the integral can be used to define an antiderivative of a continuous function. In this paper, we will present a method for establishing the existence of antiderivatives of continuous functions without using any integration theory. In addition, we will explore the potentially counter-intuitive topic of derivatives which are not Riemann integrable. It is easy to find a function whose derivative is unbounded, and thus not Riemann integrable; what is more surprising is that even bounded derivatives are not necessarily Riemann integrable. We will present two classes of functions, one conceived by Volterra and one by Pompeiu, which are differentiable on closed intervals, and whose derivatives are not Riemann integrable. Finally, we will develop the Henstock integral as a tool which integrates all derivatives.