Real Function of Two Variables/Examples/Root of Minus (x^2 + y^2)

Examples of Real Functions of Two Variables
Let $z$ denote the function defined as:
 * $z = \sqrt {-\paren {x^2 + y^2} }$

The domain of $z$ is:
 * $\Dom z = \tuple {0, 0}$

That is, just one single point, the origin of the Cartesian plane

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:
 * $\forall \tuple {x, y} \in S: \map z {x, y} = \sqrt {-\paren {x^2 + y^2} }$

From Domain of Real Square Root Function, in order for the real square root function to be defined, its argument must be non-negative.

Hence for $z$ to be defined, it is necessary for:

But:
 * $\forall x, y \in \R: x^2 + y^2 \ge 0$

Hence the only point which fulfils the condition is $\tuple {0, 0}$.

Hence the result.