Continuous Mapping is Continuous on Induced Topological Spaces

Theorem
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $\tau_{d_1}$ and $\tau_{d_2}$ be the topologies induced by the metrics $d_1$ and $d_2$.

Let $T_1 = \left({A_1, \tau_{d_1}}\right)$ and $T_2 = \left({A_2, \tau_{d_2}}\right)$ be the resulting topological spaces.

Let $f: A_1 \to A_2$ be $\left({d_1, d_2}\right)$-continuous.

Then $f$ is also $\left({\tau_{d_1}, \tau_{d_2}}\right)$-continuous.

Proof
Follows directly from:
 * the open set definition of continuity on a metric space
 * the definition of continuity on a topological space.