Page "Lipschitz continuity" Paragraph 37

* Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous.

More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set.

The Arzelà – Ascoli theorem implies that if is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence.

By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant.

In particular the set of all real-valued Lipschitz functions on a compact metric space X having Lipschitz constant ≤ K is a locally compact convex subset of the Banach space C ( X ).