Points of upper and lower semicontinuity of multivalued functions ..................

Abstract

We investigate joint upper and lower semicontinuity of two-variable set-valued functions. More precisely. among other results, we show that, under certain conditions, a two-variable lower horizontally quasicontinuous mapping $F : X \times Y \rightarrow \scr K (Z)$ is jointly upper semicontinuous on sets of the from $D \times \{ y_0\}$, where $D$ is a dense G\delta subset of $X$ and $y_0 \in Y$. A similar result is obtained for the joint lower semicontinuity of upper horizontally quasicontinuous mappings. These results improve some known results on the joint continuity of single-valued functions.