Functions represented by Rademacher series

A series of the form Σm=iVm(ί), where {a m } is a sequence of real numbers and rjt) denotes the mth Rademacher function, sign sin(2 m πέ), is called a Rademacher series (as usual, sign 0 = 0).Letting /(£) denote the sum of this series whenever it exists, we shall investigate the effect that various conditions on {a m } have on the continuity, variation, and differentiability properties of /. 2* Continuity properties* We now proveTHEOREM (2.1).// Σ I a m I < °°> then f(t) is continuous at dyadic irrationals (i.e., numbers not of the form p/2 k ) and has right and left hand limits everywhere in [0,1].