Compact sets of continuity for Borel 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 } , where K ( X ) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C ( f ) is analytic, then f is Borel; and assuming Δ 2 1 -determinacy we show that f is Borel if and only if C ( f ) is coanalytic. Similar results for projective classes are also presented.