Preimage of Subset under Mapping/Examples/Preimages of f(x, y) = (x^2 + y^2, x y)/Continuity

Example of Preimage of Subset under Mapping

 * Example-x2+y2-and-xy.png

Let $g: \R^2 \to \R^2$ be the mapping defined as:
 * $\forall \tuple {x, y} \in \R^2: \map g {x, y} = \tuple {x^2 + y^2, x y}$

$g$ is a continuous mapping.

Proof
By Continuous Mapping to Product Space it is sufficient to demonstrate that:
 * $h = \pr_1 \circ g$

and:
 * $k = \pr_2 \circ g$

where $\pr_1$ and $\pr_2$ are the first projection and seconod projection respectively on $\R^2$, are continuous.

We have:
 * $\map k {x, y} = x y$

so by the Product Rule for Continuous Real Functions, $k$ is continuous.

We have:
 * $\map h {x, y} = x^2 + y^2$

so by the Product Rule for Continuous Real Functions and the Sum Rule for Continuous Real Functions, $h$ is continuous.