Can we extend the definition of a continuous function to binary relations?

Solution 1:

There are two notions of preimage for general relations used in topology. I use the notation $y\in\phi(x)$ for $(x,y)\in\phi$. Let $\phi$ be a relation between $X$ and $Y$, aka a subset of $X\times Y$. Let $B\subseteq Y$. The upper inverse $\phi^+(B)$ of $B$ is $$\phi^+(B)=\{x\in X:\phi(x)\subseteq B\}.$$ The lower inverse $\phi^-(B)$ of $B$ is $$\phi^-(B)=\{x\in X:\phi(x)\cap B\neq\emptyset\}.$$ If $X$ and $Y$ come endowed with a topology, then we say that $\phi$ is upper hemicontinuous if the upper inverse of every open set is open, lower hemicontinuous if the lower inverse of every open set is open, and continuous if it is both upper and lower hemicontinuous.

These notions are very, very useful. Examples where continuity of relations (usually known as correspondences in this context) matters are the Maximum theorem of Berge and the Kakutani fixed point theorem. Both are fundamental tools in mathematical economics, where these notions play a big role. A great reference for these (and many other) concepts is Infinite Dimensional Analysis by Aliprantis and Border.