upper hemicontinuity

Definition

Given $A \subset \mathbb{R}^n$, the closed set $Y \subset \mathbb{R}^k$, the correspondence $f: A \to Y$ is upper hemicontinuous (u.h.c.) if it has a closed graph and the images of compact sets are bounded, i.e. for every compact $B \subset A$, the set $f(B) = \{y \in Y: y \in f(x) \text{ for some } x \in B\}$ is bounded.

Notes

A more general form exists in upper semicontinuous functions dealing with relations or correspondences in general

$f$ is a set-valued function here

See also

• lower hemicontinuity

References

1. https://math.stackexchange.com/questions/755980/the-difference-between-semicontinuity-and-hemicontinuity