Family of Lipschitz Continuous Functions with same Lipschitz Constant is Uniformly Equicontinuous

Theorem

Let $\struct {X, d}$ and $\struct {Y, d'}$ be metric spaces.

Let $\map \CC {X, Y}$ be the set of continuous functions $X \to Y$.

Let $\FF \subset \map \CC {X, Y}$ be a set of Lipschitz continuous functions all with Lipschitz constant $M \ge 0$.

Note that for each $f \in \FF$, we have:

$\map {d'} {\map f x, \map f y} \le M \map d {x, y}$

for each $x, y \in X$.

Note that if $M = 0$, we have:

$\map {d'} {\map f x, \map f y} = 0$

for each $f \in \FF$ and $x, y \in X$.

That is:

$\map {d'} {\map f x, \map f y} < \epsilon$

for each $\epsilon > 0$, $f \in \FF$ and $x, y \in X$.

So $\FF$ is uniformly equicontinuous in this case.

Suppose now that $M > 0$.

Note that whenever:

$\ds \map d {x, y} < \frac \epsilon M$

we have:

$\ds \map {d'} {\map f x, \map f y}$

$\le$

$\ds M \map d {x, y}$

$\ds $

$<$

$\ds M \times \frac \epsilon M$

$\ds $

$=$

$\ds \epsilon$

for each $f \in \FF$.

$\blacksquare$