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 x Y -> 2(Z) is strongly upper (resp. lower) quasi-continuous provided that F-J 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) is an element of Y, there is a dense G(delta) subset D of X such that F is upper (resp. lower) semi-continuous at each point of D X {y(0)}. (C) 2014 Elsevier B.V. All rights reserved.