Continuous_Mapping_is_Continuous_on_Induced_Topological_Spaces

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