Smith–Volterra–Cantor set S has positive Lebesgue measure, this means that V ′ is discontinuous on a set of positive measure. By Lebesgue's criterion for Riemann integrability, V ′ is not Riemann integrable. If one were to repeat the construction of Volterra's function with the ordinary measure-0 Cantor set C in place of the "fat" (positive-measure) Cantor set S, one would obtain a function with many similar properties, but the derivative would then be discontinuous on the measure-0 set C instead of the positive-measure set S, and so the resulting function would have a Riemann integrable derivative. See also Fundamental theorem of calculus External links Wrestling with the Fundamental Theorem of Calculus: Volterra's function, talk by David Marius Bressoud Volterra's example of a derivative that is not integrable (PPT), talk by David Marius Bressoud Fractals Measure theory General topology