Difference functions for functions with the Baire property

Let F be a family of functions f : double-struck G sign → ℝ with the Baire property, where double-struck G sign = ℝ or double-struck G sign = ℝ/ℤ. Let G ⊂ F be fixed. We continue investigatons of Keleti concerning the nature of sets H ⊂ double-struck G sign for which a function f ∈ F with the difference functions Δhf (where Δhf(x) = f(x + h) - f(x)) belonging to G for each h ∈ H, is also in G. We obtain the category analogues of Keleti's results connected with various classes of measurable functions. In particular, we consider, as G, the families of essentially (in the category sense) continuous and essentially bounded functions on double-struck G sign. We also introduce the weak difference property (in the category sense) of functions, which leads to some open problems. © Birkhäuser Verlag, Basel, 1999.