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.
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.