closed graph

Definition

Given $A \subset \mathbb{R}^N$, the closed set $Y \subset \mathbb{R}^K$, the correspondence $f: A \to Y$ has a closed graph if for any 2 sequence $x^m \to x \in A$ 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

• $f$ is a set-valued function here
• this describes a relationship between two spaces