Exercise 28, § 4

General Topology, IX

Bourbaki marks this one as harder than the rest.

Let $X$ be a metric space, let $d$ be the metric on $X$ , let $A$ be a closed subset of $X$ and let $f : A \to [1, 2]$ be a continuous real-valued function. Show that the real-valued function $g$ on $X$ which is such that $g(x) = f(x)$ if $x \in A$ , and

g(x) = \frac{1}{d(x, A)} \inf_{y \in A} (f(y)d(x, y))

if $x \in C_A$ , is continuous on $X$ . Hence give a proof of Theorem 2 for metric spaces.

Solution

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