Abstract
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→2Z is strongly upper (resp. lower) quasi-continuous provided that Fx is upper (resp. lower) semi-continuous and Fy 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∈Y, there is a dense Gδ subset D of X such that F is upper (resp. lower) semi-continuous at each point of D×{y0}.
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