Posted in Analysis and Optimization

Lipschitz continuity

In this note, we talk about the Lipschitz continuity of a function on a metric space. We followed wikipedia.

Def. Given two metric space X, Y. A function f: X \to Y is called Lipschitz continuous if there exists a real constant K \ge 0 such that for all x_1, x_2 \in X, d(f(x_1), f(x_2)) \le K d(x_1, x_2).

The smallest constant is sometimes called the (best) Lipschitz constant.

A function is called locally Lipschitz continuous if for every x \in X, there exists a neighborhood U of x such that f restricted ti U 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 X is said to be Holder continuous or to satisfy a Holder condition of order \alpha > 0 on X if there exists a constant M > 0 such that d(f(x), f(y)) \le M d(x, y)^{\alpha}.