Upper hemicontinuity

Definition

Given $A \subset \mathbb{R}^n$, the closed set $Y < \mathbb{R}^k$, the correspondence $f: A \to Y$ has a closed graph if for any 2 sequences $x^m \to x \in A$ and $y^m \to y$ with $x^m \in A$ and $y^m \in f(x^m)$ for every $m$, we have $y \in f(x)$.

Notes

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

See also

• Lower hemicontinuity

References

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