Definition:Continuity/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 $$\R$$.