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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?


migration rejected from Oct 23 '13 at 23:25

This question came from our site for professional mathematicians. Votes, comments, and answers are locked due to the question being closed here, but it may be eligible for editing and reopening on the site where it originated.

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Nate Eldredge, azimut, Stefan4024, Dan Rust, Vedran Šego
If this question can be reworded to fit the rules in the help center, please edit the question.

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

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