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.