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 $$\vartheta_{\left({A_1, d_1}\right)}$$ and $$\vartheta_{\left({A_2, d_2}\right)}$$ be the topologies induced by the metrics $$d_1$$ and $$d_2$$.

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

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

Then $$f$$ is also $\left({\vartheta_{\left({A_1, d_1}\right)}, \vartheta_{\left({A_2, d_2}\right)}}\right)$-continuous.

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