 Upper hemicontinuity

A set-valued mapping $F:X\rightrightarrows Y$ is upper hemicontinuous at $x$ when every open set $U$ containing $F(x)$ also contains $F(x^{\prime })$ for all $x^{\prime }$ in some neighborhood of $x$. Empty values are permitted. For a single-valued map this is the usual neighborhood definition of continuity.

If $Y$ is compact and the graph of a set-valued mapping is closed, upper hemicontinuity follows without a sequential-countability assumption. For each $y\in Y\setminus U$, the point $(x,y)$ is outside the closed graph, so there is a product neighborhood $N_y\times V_y$ disjoint from it. Finitely many $V_y$ cover the compact set $Y\setminus U$. Intersect their $N_y$: every nearby value is then contained in $U$.

Compact control is essential for this implication. The graph of $f(x)=1/x$ for $x\ne 0$, $f(0)=0$, is closed in ${\mathbb{R}}^2$, but the corresponding singleton-valued map is not upper hemicontinuous at zero. Its nearby outputs escape every bounded neighborhood of zero.