On the Complexity of Continuous Functions Differentiable on Cocountable Sets

Abstract

We prove that the set of all functions in $C[0,1]$, with countably many points at which the derivative does not exist, is ${\pmb \Pi}^1_1$--complete, in particular non--Borel. We obtain the classical Mazurkiewicz's theorem and the recent result of Sofronidis as corollaries from our result.