Countable compacta of continuity for projective functions

We investigate connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set C ˜ ( f ) = { K ∈ K ( X ) ; f ↾ K is continuous and K has exactly one limit point } , where K ( X ) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that C ˜ ( f ) is Borel if and only if f is Borel. Similar results for projective classes are also presented.