Let $(X,d)$ be a metric space and $\emptyset \neq A \subset X$. Show that $f(x) = d(x,A) = \inf\{d(x,a) : a \in A\}$ is a continuous function.

Question

Let $(X,d)$ be a metric space and $\emptyset \neq A \subset X$. Show that $f(x) = d(x,A) = \inf\{d(x,a) : a \in A\}$ is a continuous function.

I found this question answered on this website already using the epsilon of room technique. I was wondering if we could prove this using the fact that for continuous functions on metric spaces we have that $(x_i) \rightarrow x$ implies $f(x_i) \rightarrow f(x) \iff f$ is continuous.

Help would be greatly appreciated! Thanks!

Answer 1

Hint: Use the triangle inequality to show that $$|d(x,A)-d(y,A)|\le d(x,y),$$ then you can easily show continuity via sequences. (In fact, the above inequality means that $f$ is even Lipschitz-continuous.)