Definition:Continuous on a Metric Subspace

Definition

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ 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, $\left({Y, d_Y}\right)$ is a metric subspace of $A_1$.

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

Then $f$ is $\left({d_Y, d_2}\right)$-continuous iff:

$\forall \epsilon > 0: \exists \delta > 0: d_Y \left({x, a_1}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a_1}\right)}\right) < \epsilon$

Warning

Note that a function which is $\left({d_Y, d_2}\right)$-continuous might not also be $\left({d_1, d_2}\right)$-continuous.

For example, let $f: \R \to \R$ be given by:

$f \left({x}\right) = \begin{cases} 0 & : x \in \Q \\ 1 & : x \in \R \end{cases}$

where $\Q$ is the set of rational numbers.

Then $f \restriction_{\Q}: \Q \to \R$ is the constant function $f_0$ with value $0$, which is continuous at every point, but $f$ is not continuous on $\R$.