Example 2

Let X, Y be two metric spaces, d (resp. $d'$) the metric on X (resp. Y), and let k, $\alpha$ be two real numbers > 0. Then the set of all mappings $f : X \to Y$ such that

for each pair of points $x, x'$ of X, is uniformly equicontinuous. For example, the set of all isometries (Chapter IX, § 2, no. 2) of X onto a subset of Y is uniformly equicontinuous.

\* Let H be a set of real-valued functions defined on an interval $I \subset \mathbf{R}$, which are differentiable on I and are such that $|f'(x)| \leq k$ for all $x \in I$ and all $f \in H$. Then H is uniformly equicontinuous, because if $x_1, x_2$ are any two points of I we have $|f(x_1) - f(x_2)| \leq k|x_1 - x_2|$ for each $f \in H$ by the mean value theorem. \*