Definition:Continuous Mapping (Metric Space)/Space/Definition 2

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 $A_2$.

$f$ is continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$ :
 * for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \sqbrk U$ is open in $M_1$.

By definition, this is equivalent to the continuity of $f$ with respect to the induced topologies on $A_1$ and $A_2$.

Also see

 * Equivalence of Definitions of Continuity on Metric Spaces