Vito Volterra's construction of a nonconstant function with a bounded, non-Riemann integrable derivative

Abstract

In the 1880s the research on the theory of integration was focused mainly on the properties of infinite sets. The development of nowhere dense sets with positive outer content, known nowadays as nowhere dense sets with positive measure, allowed the construction of general functions with the purpose of extending Riemann's definition of integral. Vito Volterra provided, in 1881, an example of a differentiable function whose derivative is bounded but not Riemann integrable. In this article we present and discuss Volterra's example.