# Continuity of functions between Metric Spaces [closed]

Prove that every function mapping $(X,d_0)$ to a metric space $(Y,d)$ is continuous.

where $d_0$ is the discrete metric.How do we prove this?

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## closed as off-topic by Nate Eldredge, azimut, Stefan4024, Dan Rust, Vedran ŠegoOct 23 '13 at 23:25

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What have you tried. Where are you having trouble with proving this? Show us some work :) – Luiz Cordeiro Oct 23 '13 at 22:04
You need to show that if $U$ is an open subset of $Y$, then $f^{-1}[U]$ is an open subset of $X$. Can you tell me what subsets of $X$ are open in the discrete metric? If you can, the problem is trivial. If not, that’s where you should begin thinking about it. – Brian M. Scott Oct 23 '13 at 22:06