Let $(X,d)$ be a metric space, and suppose $A \subseteq X$ is nonempty. Define $$d(x,A) = \inf_{a\in A} (d(x,a) )$$ and let $f: X \rightarrow \mathbb{R}$ be given by $x \mapsto d(x,A)$. How do I show that $f$ is continuous? I have tried the open ball definitions, and the $\epsilon, \delta$ definitions, both without much success. This problem comes from Introduction to Topology by Mendelson.
Showing that $f: X \rightarrow \mathbb{R}$ where $x \mapsto d(x,A)$ is contineous in a metric space.
Asked
Active
Viewed 81 times
0
-
2You can show that it is $1$-Lipschitz thanks to the triangle inequality – Didier Oct 14 '23 at 15:12
-
This is a dulicate and has been asked before. Ask for why distance is continuous – FShrike Oct 14 '23 at 22:49
-
@FShrike I am not sure what you mean by distance is continuous. Do you mean ask why $d: X\times X \rightarrow \mathbb{R}$ is continuous? – Mani Oct 15 '23 at 00:29
-
https://math.stackexchange.com/q/764103 – FShrike Oct 15 '23 at 00:30