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→2ZF:X×Y→2Z is strongly upper (resp. lower) quasi-continuous provided that FxFx is upper (resp. lower) semi-continuous and FyFy is lower (resp. upper) quasi-continuous. Moreover, we will prove that if F is compact-valued and Z   is second countable, then for each y0∈Yy0∈Y, there is a dense GδGδ subset D of X such that F   is upper (resp. lower) semi-continuous at each point of D×{y0}D×{y0}.