I'm studying the proof of the following theorem, but there are two things that I have not been able to understand about the continuity of the function $g$ on the boundary of $A$.
(1) Why $\inf_{y \in A}(f(y)d(x',y)) = \inf_{y\in C}(f(y)d(x',y))$?
(2) How do I deduce that $d(x',A) = d(x',C) = \inf_{y\in C}d(x',y)$ and how does this imply that $(f(x)-\varepsilon)d(x',A) \leq \inf_{y \in A}(f(y)d(x',y)) \le (f(x)+\varepsilon)d(x',A)$?
I appreciate any explanation.
