McShane–Whitney extension

Remarks

Proof idea: fix $a_0 \in A$; for $a \in A$, $f(a) + L d(x, a) \ge f(a_0) - L d(a, a_0) + L d(x, a) \ge f(a_0) - L d(x, a_0)$, so the infimum is finite. For $x \in A$ the same inequality with $a_0 = x$ shows $F(x) \ge f(x)$, and the choice $a = x$ gives $F(x) \le f(x)$. Each function $x \mapsto f(a) + L d(x, a)$ is $L$-Lipschitz, and an infimum of $L$-Lipschitz functions that is finite is $L$-Lipschitz. Example: on $X = \R$ with $A = \set{0, 1}$, $f(0) = 0$, $f(1) = 1$ and $L = 1$, both $F$ and $G$ equal $x$ on $[0, 1]$, while $F(x) = \abs{x}$ and $G(x) = x$ for $x \le 0$, and $F(x) = x$ and $G(x) = 1 - \abs{x - 1}$ for $x \ge 1$. For maps into $\R^m$ with $m \ge 2$, the same constant needs the Kirszbraun theorem.

Going deeper

For a student

The result needs no structure on $X$ beyond the metric, unlike the vector-valued Kirszbraun theorem. It shows, for example, that every Lipschitz function on a closed set of $\R^n$ extends to $\R^n$, which is useful before applying Rademacher's theorem.