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.
Let $x_1,x_2\in X$, and $y\in Y$. We have $d(x_1,y)\leq d(x_1,x_2)+d(x_2,y)$ and taking the infimum over $y\in Y$ we get $f(x_1)\leq d(x_1,x_2)+f(x_2)$ so $f(x_1)-f(x_2)\leq d(x_1,x_2)$ and by symmetry $|f(x_1)-f(x_2)|\leq d(x_1,x_2)$ so $f$ is $1$-Lipschitz.
If $x$ is in the closure of $Y$ then $\{y,d(x,y)