We show that it is consistent with ZFC that the family of functions with the Baire property has the difference property. That
is, every function for which f(x + h)-f(x) has the Baire property for every h∈R is of the form f=g + Awhere g has the Baire property and A is additive.
The connection between transitivity and existence of a dense orbit for multifunctions in generalized topological spaces is studied. Moreover strongly transitive multifunctions and functions in generalized topological spaces are investigated.
We study sets of points at which ω1 sequences of real functions from a given class F converge. As F we consider continuous functions, first class of Baire, Borel
measurable functions, functions with Baire property and Lebesgue measurable functions. Connections of those problem with additional
set-theoretic axioms are discussed.