Definition 9

A graph F is said to be a functional graph if for each x there is at most one object which corresponds to x under F (Chapter I, § 5, no. 3). A correspondence $f = (\mathrm{F}, \mathrm{A}, \mathrm{B})$ is said to be a function if its graph F is a functional graph and if its source A is equal to its domain $\mathrm{pr}_1\mathrm{F}$. In other words, a correspondence $f = (\mathrm{F}, \mathrm{A}, \mathrm{B})$ is a function if for every x belonging to the source A of f the relation $(x, y) \in \mathrm{F}$ is functional in y (Chapter I, § 5, no. 3); the unique object which corresponds to x under f is called the value of f at the element x of A, and is denoted by $f(x)$ (or $f_x$, or $\mathrm{F}(x)$, or $\mathrm{F}_x$).

If $f$ is a function, F its graph, and $x$ an element of the domain of $f$, the relation $y = f(x)$ is then equivalent to $(x, y) \in \mathrm{F}$ (Chapter I, § 5, no. 3, criterion C46).