Definition:Continuous Mapping (Metric Space)

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 $a \in A_1$ be a point in $A_1$.

Equivalence of Definitions
All these statements are equivalent by Equivalence of Metric Space Continuity Definitions.

Also known as
A mapping which is continuous from $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$ can also be referred to as $\left({d_1, d_2}\right)$-continuous.

Also see

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


 * Image of Open Set under Continuous Mapping in Metric Space may not be Open