Characterization of quasi-continuity of multifunctions of two variables (Q2810990)

The authors investigate the relationship between separate and joint quasi-continuity of a multifunction with two variables, continuing the previous work of \textit{S. Kempisty} [Fundam. Math. 19, 184--197 (1932; Zbl 0005.19802)] and [\textit{T. Neubrunn}, Math. Slovaca 32, 147--154 (1982; Zbl 0483.54009)]. The main result of this paper is a theorem stating the necessary and sufficient conditions for a multifunction with two variables to be jointly quasi-continuous. It says that a compact-valued multifunction \(F\: X \times Y \rightarrow Z\), where \(X\) is a Baire space, \(Y\) is a second countable space and \(Z\) is a separable metrizable space, is quasi-continuous if and only if \(F\) is horizontally quasi-continuous and there exists a residual subset \(M\) of \(X\) such that for any \(x \in M\) the multifunction \(F^x = F(x, \cdot)\) is quasi-continuous on \(Y\).