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

Theorem
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 $A_2$.

Then $f$ is $\left({d_1, d_2}\right)$-continuous iff:


 * for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \left({U}\right)$ is open in $M_1$.

Also used as definition
Some treatments of this subject use this property of continuity to define the concept.

Also see

 * Metric Space Continuity by Epsilon-Delta
 * Metric Space Continuity by Open Ball