Real Function of Two Variables/Examples/x + y

Examples of Real Functions of Two Variables
Let $z$ denote the function defined as:
 * $z = x + y$

The domain of $z$ is:
 * $\Dom z = \R \times \R$

Proof
The domain of $z$ is given implicitly and conventionally.

What is meant is:
 * $z: S \to \R$ is the function defined on the largest possible subset $S$ of $\R^2$ such that:
 * $\forall \tuple {x, y} \in S: \map z {x, y} = x + y$

There are no restrictions on either $x$ or $y$ for $\map z {x, y}$ to be defined.

Hence the domain of $z$ is the entire Cartesian plane.