Let F be a family of functions f: G-->R with the Baire property, where G=R or G=R/Z. Let G be a fixed subset of F. We continue investigatons of Keleti concerning the nature of sunsets H of G for which a function f in F, with the difference functions \Delta_h{f} (where \Delta_h{f}(x)=f(x+h) - f(x)) belonging to G for each h in H, is also in G. We obtain the category analogs 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 G. We also introduce the weak difference property (in the category sense) of functions, which leads to some open problems.