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Let $(X,d)$ be a metric space and $Y\subset X$ be a non-empty subset. Is the map given by$$f(x)=\inf\lbrace d(x,y)\colon y\in Y\rbrace$$ a Lipschitz map? And does the equivalence $f(x)=0\iff x\in$ closure$(Y)$ hold?

Thank you in advance.

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    [yes it is Lipschitz](http://math.stackexchange.com/a/48853/5363) and yes the equivalence holds for $\Leftarrow$ use continuity and for $\Rightarrow$ use the definition of closure by sequences.2012-03-19

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