What properties does $R$ have to satisfy for $R=\operatorname{graph}(f)$ for some function $f:A\to B$?

Question

Given a relation $R$ between $A$ and $B$, what properties does $R$ have to satisfy for $R=\operatorname{graph}(f)$ for some function $f:A\to B$?

Answer 1

You need two properties.

1) For each $x\in A$ there is a $y\in B$ such that $(x,y)\in R$

2) If $(x_1,y_1)\in R$ and $(x_2,y_2)\in R$, then $$x_1 = x_2 \implies y_1=y_2$$