Sets of everywhere singular functions

Abstract

In this paper we present a simple general method for demonstrating that in certain function spaces various sets consisting of functions that exhibit at every point a prescribed kind of singularity form a coanalytic but not Borel set. We illustrate this method by providing new proofs that the set of nowhere differential continuous functions on [0,1] is (coanalytic but) not Borel and similarly for the set of continuous functions on [0,1] which fail everywhere to have a unilateral derivative (including ± ∞) -- the so called Besicovitch functions. These results were originally proved by Mauldin, [Mau] and unpublished, respectively. We also give a new example of a coanalytic not Borel set, namely the set of integrable functions with everywhere divergent Fourier series. Finally, we formulate an abstract theorem, which includes as simple instances all the above and other similar examples.