Real Function of Two Variables/Examples/x + y
Jump to navigation
Jump to search
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.
$\blacksquare$
Sources
- 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text C$: Function of Two Independent Variables: Example $2.62: 2$