Continuous Mapping (Metric Space)/Examples/Composition of Arbitrary Mappings
Examples of Continuous Mappings in the Context of Metric Spaces
Let the following mappings be defined:
\(\ds g: \R^2 \to \R^2 \times \R^2: \, \) | \(\ds \map g {x, y}\) | \(=\) | \(\ds \tuple {\tuple {x, y}, \tuple {x, y} }\) | |||||||||||
\(\ds h: \R^2 \times \R^2 \to \R \times \R: \, \) | \(\ds \map h {\tuple {a, b}, \tuple {c, d} }\) | \(=\) | \(\ds \tuple {a + b, c - d}\) | |||||||||||
\(\ds k: \R \times \R \to \R \times \R: \, \) | \(\ds \map k {u, v}\) | \(=\) | \(\ds \tuple {u^2, v^2}\) | |||||||||||
\(\ds m: \R \times \R \to \R: \, \) | \(\ds \map k {x, y}\) | \(=\) | \(\ds \dfrac {x - y} 4\) |
where $\R$ and $\R^2$ denote the real number line and real number plane respectively, under the usual (Euclidean) metric.
Then:
- each of $g, h, k, m$ are continuous
- $x y = \map {\paren {m \circ k \circ h \circ g} } {x, y}$
where $\circ$ denotes composition of mappings.
Proof
By Composition of Mappings is Associative and General Associativity Theorem, $m \circ j \circ h \circ g$ is well-defined and unambiguous.
Let $d : \R^2 \times \R^2 \to \R$ be defined as:
- $\map d {\tuple {a, b}, \tuple {c, d} } = \max \set {\size {a - c}, \size {b - d} }$
and $d': \paren {\R^2 \times \R^2} \times \paren {\R^2 \times \R^2} \to \R$ be defined as:
\(\ds \map {d'} {\tuple {\tuple {a, b}, \tuple {c, d} }, \tuple {\tuple {k, l}, \tuple {m, n} } }\) | \(=\) | \(\ds \max \set {d {\tuple {a, b}, \tuple {k, l} }, d {\tuple {c, d}, \tuple {m, n} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\size {a - k}, \size {b - l}, \size {c - m}, \size {d - n} }\) |
Let $\epsilon \in \R_{>0}$.
Let $d$ be constrained by some $\delta \in \R_{>0}$ such that $\delta < \epsilon$:
- $\map d {\tuple {x, y}, \tuple {a, b} } < \delta$
and so:
- $\size {x - a} <\delta$ and $\size {y - b} < \delta$
Then:
\(\ds \map {d'} {\map g {x, y}, \map g {a, b} }\) | \(=\) | \(\ds \max \set {\map d {\tuple {x, y}, \tuple {a, b} }, \map d {\tuple {x, y}, \tuple {a, b} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map d {\tuple {x, y}, \tuple {a, b} }\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \delta\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) |
Hence:
- $\map d {x, a} < \delta \implies \map {d'} {\map g x, \map g a} < \epsilon$
where $x, a \in \R^2$.
Hence $g$ is continuous.
$\Box$
We have that:
- $h: \R^2 \times \R^2 \to \R \times \R: \map h {\tuple {a, b}, \tuple {c, d} } = \tuple {a + b, c - d}$
Let $d$ be defined as elements of $\R^2 \times \R^2$ as above.
Let $\epsilon \in \R_{>0}$.
Let $d$ be constrained by some $\delta \in \R_{>0}$ such that $\delta < \dfrac \epsilon 2$:
- $\map d {\tuple {\tuple {a, b}, \tuple {c, d} }, \tuple {\tuple {k, l}, \tuple {m, n} } } = \max \set {\size {a - k}, \size {b - l}, \size {c - m}, \size {d - n} } < \delta$
Then:
\(\ds \) | \(\) | \(\ds \map {d'} {\map h {\tuple {a, b}, \tuple {c, d} }, \map h {\tuple {k, l}, \tuple {m, n} } }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {d'} {\tuple {a + b, c - d}, \tuple {k + l, m - n} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \max \set {\size {\size {a - k} + \size {b - l} }, \size {\size {c - m} + \size {d - n} } }\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds 2 \delta\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) |
Hence:
- $\map d {x, a} < \delta \implies \map {d'} {\map h x, \map h a} < \epsilon$
where $x, a \in \R^2 \times \R^2$.
Hence $h$ is continuous.
$\Box$
We have that:
- $k: \R \times \R \to \R \times \R: \map k {u, v} = \tuple {u^2, v^2}$
Let $d$ be defined as:
\(\ds \map d {\tuple {u, v}, \tuple {a, b} }\) | \(=\) | \(\ds \max \set {\size {u - a}, \size {v - b} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d'} {\map k {u, v}, \map k {a, b} }\) | \(=\) | \(\ds \max \set {\size {u^2 - a^2}, \size {v^2 - b^2} }\) |
Without loss of generality, let it be assumed that $\size {u - a} \ge \size {v - b}$.
Let $u > a$.
Then:
- $\size {u - a} = u - a$
Setting $\map d {\tuple {u, v}, \tuple {a, b} } < \delta$ gives:
- $u - a < \delta$
and so:
- $u < a + \delta$
Hence:
- $\size {u^2 - a^2} < \size {\delta^2 + 2 a \delta}$
Setting $\epsilon = \size {\delta^2 + 2 a \delta}$, it is noted that this can be made arbitrarily small for any given finite $a$.
Also:
- $\size {v - b} < \delta$
and so:
- $\size {v^2 - b^2} < \size {\delta^2 + 2 b \delta}$
As all the elements are even, $a > u$ gives similar results.
By the same argument, $\size {v - b} \ge \size {u - a}$ gives the same result.
So in all cases:
- $\map d {\tuple {u, v}, \tuple {a, b} } < \delta \implies \map {d'} {\map k {u, v}, \map l {a, b} } < \epsilon$
for whatever $\epsilon$ is chosen.
Hence $k$ is continuous.
$\Box$
We have that:
- $m: \R \times \R \to \R: \map k {x, y} = \dfrac {x - y} 4$
Let $d$ be defined as:
\(\ds \map d {\tuple {x, y}, \tuple {a, b} }\) | \(=\) | \(\ds \max \set {\size {x - a}, \size {y - b} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d'} {\map m {x, y}, \map m {a, b} }\) | \(=\) | \(\ds \size {\dfrac {x - y} 4 - \dfrac {a - b} 4}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 1 4 \size {x + b - y - a}\) | ||||||||||||
\(\ds \) | \(<\) | \(\ds \size {x + b - y - a}\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \size {\size {x - a} + \size {y - b} }\) |
Let $\epsilon \in \R_{>0}$.
Let $d$ be constrained by some $\delta \in \R_{>0}$ such that $\delta < \dfrac \epsilon 2$:
\(\ds \map d {\tuple {x, y}, \tuple {a, b} }\) | \(<\) | \(\ds \delta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \size {x - a}\) | \(<\) | \(\ds \delta \land \size {y - b} < \delta\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d'} {\map m {x, y}, \map m {a, b} }\) | \(<\) | \(\ds 2 \delta\) | |||||||||||
\(\ds \) | \(<\) | \(\ds \epsilon\) |
Hence:
- $\map d {\tuple {x, y}, \tuple {a, b} } < \delta \implies \map {d'} {\map m {x, y}, \map m {a, b} } < \epsilon$
Hence $m$ is continuous.
$\Box$
Consider $\tuple {x, y} \in \R^2$.
We have:
\(\ds \map {\paren {m \circ k \circ h \circ g} } {x, y}\) | \(=\) | \(\ds \map {\paren {m \circ k \circ h} } {\map g {x, y} }\) | Definition of Composition of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {m \circ k \circ h} } {\tuple {x, y}, \tuple {x, y} }\) | Definition of $g$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {m \circ k} } {\map h {\tuple {x, y}, \tuple {x, y} } }\) | Definition of Composition of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {\paren {m \circ k} } {x + y, x - y}\) | Definition of $h$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map m {\map k {x + y, x - y} }\) | Definition of Composition of Mappings | |||||||||||
\(\ds \) | \(=\) | \(\ds \map m {\paren {x + y}^2, \paren {x - y}^2}\) | Definition of $k$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map m {x^2 + 2 x y + y^2, x^2 - 2 x y + y^2}\) | expanding | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\paren {x^2 + 2 x y + y^2} - \paren {x^2 - 2 x y + y^2} } 4\) | Definition of $m$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x y\) | simplification |
$\blacksquare$
Sources
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 3$: Continuity: Exercise $4$