Continuity Points of Typical Bounded Functions

The study of typical continuous functions has been one of the most popular topics in classical real analysis. Several people have also investigated typical behaviour in other families of functions, such as those of bounded functions of Baire class 1, of bounded Darboux functions of Baire class 1, and of bounded derivatives, where the topology is given by the supremum norm. Kostyrko and Šalát [2] proved a theorem, applicable to general families of bounded functions, on the continuity points of typical functions. They showed that if a linear space of bounded functions has an element that is discontinuous almost everywhere, then a typical element in the space is discontinuous almost everywhere (what they actually proved is slightly stronger than this; see below for the precise statement). Our aim in this paper is to give a topological analogue of this theorem, by showing that if a linear space of bounded functions has an element that is discontinuous everywhere in a residual set, then a typical element in the space is discontinuous everywhere in a residual set (again, we in fact show a slightly stronger result). Let us fix notation and give the precise statements of the theorem of Kostyrko and Šalát and of our main theorem. By a function we shall always mean a real-valued function defined on the unit interval [0, 1]. Let b denote the Banach space of all bounded functions, equipped with the supremum norm. For a function f , we write C(f) and D(f) for the sets of all continuity and discontinuity points of f respectively. The Lebesgue measure on [0, 1] will be denoted by μ.