Strong quasi-continuity of set-valued functions

By means of topological games, we will show that under certain circumstances on topological spaces X, Y and Z, every two variable set-valued function F : X × Y → 2 Z is strongly upper (resp. lower) quasi-continuous provided that F x is upper (resp. lower) semi-continuous and F y is lower (resp. upper) quasi-continuous. Moreover, we will prove that if F is compact-valued and Z is second countable, then for each y 0 ∈ Y , there is a dense G δ subset D of X such that F is upper (resp. lower) semi-continuous at each point of D × { y 0 } .