Definition:Continuity/Metric Subspace

Definition

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 $Y \subseteq A_1$.

By definition, $\struct {Y, d_Y}$ is a metric subspace of $A_1$.

Let $a \in Y$ be a point in $Y$.

Then $f$ is $\tuple {d_Y, d_2}$-continuous at $a$ if and only if:

$\forall \epsilon > 0: \exists \delta > 0: \map {d_Y} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$

Similarly, $f$ is $\tuple {d_Y, d_2}$-continuous if and only if:

$\forall a \in Y: f$ is $\tuple {d_Y, d_2}$-continuous at $a$

Also see

• Restriction of Non-Continuous Mapping on Metric Space to Subspace may be Continuous

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Example $2.2.5$