Addition of Coordinates on Cartesian Plane under Chebyshev Distance is Continuous Function
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Theorem
Let $\R^2$ be the real number plane.
Let $d_\infty$ be the Chebyshev distance on $\R^2$.
Let $f: \R^2 \to \R$ be the real-valued function defined as:
- $\forall \tuple {x_1, x_2} \in \R^2: \map f {x_1, x_2} = x_1 + x_2$
Then $f$ is continuous.
Proof
First we note that:
\(\ds \size {\paren {x_1 + x_2} - \paren {y_1 + y_2} }\) | \(=\) | \(\ds \size {\paren {x_1 - y_1} + \paren {x_2 - y_2} }\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \size {x_1 - y_1} + \size {x_2 - y_2}\) | Triangle Inequality for Real Numbers | |||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(\le\) | \(\ds 2 \max \set {\size {x_1 - y_1}, \size {x_2 - y_2} }\) |
Let $\epsilon \in \R_{>0}$.
Let $x = \tuple {x_1, x_2} \in \R^2$.
Let $\delta = \dfrac \epsilon 2$.
Then:
\(\ds \forall y = \tuple {y_1, y_2} \in \R^2: \, \) | \(\ds \map {d_\infty} {x, y}\) | \(<\) | \(\ds \delta\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \max \set {\size {x_1 - y_1}, \size {x_2 - y_2} }\) | \(<\) | \(\ds \delta\) | Definition of Chebyshev Distance on Real Number Plane | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \max \set {\size {x_1 - y_1}, \size {x_2 - y_2} }\) | \(<\) | \(\ds \epsilon\) | Definition of $\epsilon$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {\paren {x_1 + x_2} - \paren {y_1 + y_2} }\) | \(<\) | \(\ds \epsilon\) | from $(1)$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {\map f x - \map f y}\) | \(<\) | \(\ds \epsilon\) | Definition of $f$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map d {\map f x, \map f y}\) | \(<\) | \(\ds \epsilon\) | Definition of Usual Metric on $\R$ |
Thus it has been demonstrated that:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall y \in \R^2: \map {d_\infty} {x, y} < \delta \implies \map d {\map f x, \map f y} < \epsilon$
Hence by definition of continuity at a point, $f$ is continuous at $x$.
As $x$ was chosen arbitrarily, it follows that $f$ is continuous for all $x \in \R^2$.
The result follows by definition of continuous mapping.
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 3$: Continuity: Exercise $3$