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 $d(f(x), f(y)) \le M d(x, y)^{\alpha}$. 