Metric Space Continuity by Open Ball
Theorem
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.
Let $a \in A_1$ be a point in $A_1$.
The following definitions of the concept of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:
$\epsilon$-$\delta$ Definition
$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$) if and only if:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$
where $\R_{>0}$ denotes the set of all strictly positive real numbers.
$\epsilon$-Ball Definition
$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$) if and only if:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {a; d_1} } \subseteq \map {B_\epsilon} {\map f a; d_2}$
where $\map {B_\epsilon} {\map f a; d_2}$ denotes the open $\epsilon$-ball of $\map f a$ with respect to the metric $d_2$, and similarly for $\map {B_\delta} {a; d_1}$.
Proof
$\epsilon$-$\delta$ Definition implies $\epsilon$-Ball Definition
Suppose that $f$ is $\tuple {d_1, d_2}$-continuous at $a$ in the sense that:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$
where $\R_{>0}$ denotes the set of all strictly positive real numbers.
Let $\epsilon \in \R_{>0}$ be arbitrary.
Let $\delta$ be such that:
- $\forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$
as is known to exist by hypothesis.
Then:
\(\ds y\) | \(\in\) | \(\ds f \sqbrk {\map {B_\delta} {a; d_1} }\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists x \in \map {B_\delta} {a; d_1}: \, \) | \(\ds y\) | \(=\) | \(\ds \map f x\) | Definition 1 of Image of Subset under Mapping | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_1} {x, a}\) | \(<\) | \(\ds \delta\) | Definition of Open Ball | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_2} {\map f x, \map f a}\) | \(<\) | \(\ds \epsilon\) | by hypothesis | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_2} {y, \map f a}\) | \(<\) | \(\ds \epsilon\) | as $y = \map f x$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(\in\) | \(\ds \map {B_\epsilon} {\map f a; d_2}\) | Definition of Open Ball | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds f \sqbrk {\map {B_\delta} {a; d_1} }\) | \(\subseteq\) | \(\ds \map {B_\epsilon} {\map f a; d_2}\) | Definition of Subset |
As $\epsilon$ is arbitrary, it follows that:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {a; d_1} } \subseteq \map {B_\epsilon} {\map f a; d_2}$
$\Box$
$\epsilon$-Ball Definition implies $\epsilon$-$\delta$ Definition
Suppose that $f$ is $\tuple {d_1, d_2}$-continuous at $a$ in the sense that:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {a; d_1} } \subseteq \map {B_\epsilon} {\map f a; d_2}$
where $\map {B_\epsilon} {\map f a; d_2}$ denotes the open $\epsilon$-ball of $\map f a$ with respect to the metric $d_2$, and similarly for $\map {B_\delta} {a; d_1}$.
Let $\epsilon \in \R_{>0}$ be arbitrary.
Let $\delta$ be such that:
- $f \sqbrk {\map {B_\delta} {a; d_1} } \subseteq \map {B_\epsilon} {\map f a; d_2}$
as is known to exist by hypothesis.
Then:
\(\ds \map {d_1} {x, a}\) | \(<\) | \(\ds \delta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(\in\) | \(\ds \map {B_\delta} {a; d_1}\) | Definition of Open $\epsilon$-Ball | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map f x\) | \(\in\) | \(\ds \map {B_\epsilon} {\map f a; d_2}\) | Definition 1 of Image of Subset under Mapping | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {d_2} {\map f x, \map f a}\) | \(<\) | \(\ds \epsilon\) | Definition of Open $\epsilon$-Ball |
As $\epsilon$ is arbitrary, it follows that:
- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \sqbrk {\map {B_\delta} {a; d_1} } \subseteq \map {B_\epsilon} {\map f a; d_2}$
$\blacksquare$
Also see
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: $\varepsilon$-Balls
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 4$: Open Balls and Neighborhoods: Theorem $4.2$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Definition $2.3.6$