If f : (M,) (N,) is a continuous function between two metric spaces, then f : (M, ) (N, ) is a

If f : (M,ρ) −→ (N,σ) is a continuous function between two metric spaces, then f : (M, τρ ) −→ (N, τσ) is a continuous function between the topological spaces with the topologies induced by the metrics.

how can I prove it rigorously?