In this paper, we will show that under certain circumstances, a two variable set-valued mapping $F:X \\\\\\\\\\\\\\\\times Y \\\\\\\\\\\\\\\\to 2^Z$ is jointly continuous on a set of the form $D\\\\\\\\\\\\\\\\times \\\\\\\\\\\\\\\\{y\\\\\\\\\\\\\\\\}$, where $D$ is a dense $G_\\\\\\\\\\\\\\\\delta$ subset of $X$ and $y\\\\\\\\\\\\\\\\in Y$. Among some other results, we will show that if $X$ is a Baire space, $Y$ a topological space and $Z$ is a Tychonoff locally weakly $k$-developable space, then every set-valued mapping $F$ from $ X \\\\\\\\\\\\\\\\times Y$ into compact subsets of $Z$, is Vietoris continuous at each point of $X \\\\\\\\\\\\\\\\times \\\\\\\\\\\\\\\\{y_0\\\\\\\\\\\\\\\\}$, where $y_0 \\\\\\\\\\\\\\\\in Y$ is a $W$-point, provided that all its $x$-sections $F_x$ are Vietoris continuous and all its $y$-sections $F^y$ are Vietoris quasi-continuous. Our results improve some old related results in the literature.

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