Preimage of Subset under Mapping/Examples/Preimages of f(x, y) = (x^2 + y^2, x y)/Continuity
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Example of Preimage of Subset under Mapping
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.
$\blacksquare$
Sources
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.5$: Products