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

Example of Preimage of Subset under Mapping

Yellow: $g^{-1} \sqbrk {\openint 0 3 \times \openint 0 1}$ (boundary not included)

Let $g: \R^2 \to \R^2$ be the mapping defined as:

$\forall \tuple {x, y} \in \R^2: \map g {x, y} = \tuple {x^2 + y^2, x y}$

Let the following subset of $\R^2$ be defined:

$S = g^{-1} \sqbrk {\openint 0 3 \times \openint 0 1}$

Then:

$S = \set {\tuple {x, y} \in \R^2: x y < 1} \cap \set {\tuple {x, y} \in \R^2: 0 < x^2 + y^2 < 3}$

Continuity

$g$ is a continuous mapping.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Example $2.3.16 \ \text {(b)}$