Upper hemicontinuity

A set-valued mapping $F$ is upper hemicontinuous at $x$ when every open set containing $F(x)$ also contains $F(x^{\prime })$ for all nearby $x^{\prime }$. With values in a fixed compact space, a closed graph of a set-valued mapping implies upper hemicontinuity: a violating sequence has a convergent subsequence of outputs whose limit lies in the limiting value.