Consider the metric space $(X,d)$ and the non-empty subset $A \subseteq X$. What can we say about the (uniform) continuity of the function $d_{A}: X \rightarrow \mathbb{R}: x \mapsto d_{A}(x) = \inf \{ d(a,x) | a \in A \}$? Should I make a case study on whether X is compact or not? Any hints? Thanks!
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A google search for "metric uniform continuous" led this (see the first answer): https://math.stackexchange.com/questions/8066/is-the-distance-function-in-a-metric-space-uniformly-continuous – ThePortakal Jun 17 '17 at 12:27
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@ThePortakal see also the answer below. – hamam_Abdallah Jun 17 '17 at 12:40
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hint
For $x,y \in X $ and $a\in A $, we have
$$d (a,x)\le d (a,y)+d (x,y) $$ $$\implies d_A (x)\le d (a,y)+d (x,y) $$ $$\implies d_A (x)-d (x,y)\le d (a,y) $$
$$\implies d_A (x)-d (x,y)\le d_A (y) $$ $$\implies d_A (x)-d_A (y)\le d (x,y) $$ and by symetry, $$d_A (y)-d_A (x)\le d (x,y) $$
and finally $$|d_A (x)-d_A (y)|\le d (x,y) $$ which proves that $x\mapsto d_A (x) $ is Lipschitz and uniformly continuous at $X $.
hamam_Abdallah
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