Two large subsets of a functional space

Abstract

Let P1 denote the Banach space composed of all bounded derivatives f of everywhere differentiable functions on [0,1] such that the set of points where f vanishes is dense in [0,1]. Let D0 consist of those functions in P that are unsigned on every interval, and let D1 consist of those functions in P1 that vanish on dense subsets of measure zero. Then D0 and D1 are dense Gδ-subsets of P1 with void interior. Neither D0 nor D1 is a subset of the other.