In this note, we talk about the Lipschitz continuity of a function on a metric space. We followed wikipedia.
Def. Given two metric space . A function is called Lipschitz continuous if there exists a real constant such that for all , .
The smallest constant is sometimes called the (best) Lipschitz constant.
A function is called locally Lipschitz continuous if for every , there exists a neighborhood of such that restricted ti is Lipschitz continuous. Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X. In spaces that are not locally compact, this is a necessary but not a sufficient condition.
More generally, a function f defined on is said to be Holder continuous or to satisfy a Holder condition of order on if there exists a constant such that .