Lipschitz Functions, Part 1

What is a Lipschitz Function?

This is the first post in a series of posts about Lipschitz functions. These are functions that fit right in-between the continuous functions and the differentiable functions. So, every Lipschitz function is continuous, but not necessarily differentiable. In time, we will go through the basic definitions of Lipschitz functions, how to put a Banach space structure onto the space of Lipschitz functions, and some cool uses of Lipschitz, but not differentiable, functions.

A function from $\mathbb{R}$ to $\mathbb{R}$ is called $L$-Lipschitz if it satisfies the following bound for all $x, y \in \mathbb{R}$.

If a function is called Lipschitz if it is $L$-Lipschitz for some $L$. The least such $L$ is called the Lipschitz constant of that function.

For example, take $f(x) = |x|$. This is a 1-Lipschitz function, since the slope between any two points on the graph of $f$ is at most 1. Other 1-Lipschitz functions include $f(x) = x$ and $f(x) = \sin(x)$. Lipschitzness can be interpreted as a bound on the regularity of a function, or how smooth it is. Differentiability is another condition on the regularity of a function, and it is interesting to note that $f(x) = |x|$ is Lipschitz, but is not differentiable.

Broadly, there are two ways that a function can fail to be Lipschitz: either the Lipschitz condition fails when $|x - y|$ is large, or it fails when $|x - y|$ is small. For example, $f(x) = \sqrt{|x|}$ satisfies the Lispchitz condition with $L = 1$ whenever $|x - y| \geq 1$. But, it fails when $x$ and $y$ get too close to each other. On the other hand, $f(x) = x^2$ satisfies a Lipschitz condition in any bounded domain, but doesn't satisfy it when its domain is the entire real line. Focusing only on global Lipschitzness or local Lipschitzness leads to more general theories than just the theory of Lipschitz functions. There are a few different ways to approach functions that are Lipschitz on large scales, although I don't know much about them. This set of notes is more focused on the functions which satisfy Lipschitz conditions locally — that is, on small scales.

More specifically, define a function to be locally Lipschitz if its domain can be covered by intervals $I$ such that for some constant $L$ depending on $I$, and all $x, y \in I$, the following holds.

We can see from this definition that $f(x)=x^2$ is a locally Lipschitz function, although as noted, it is not globally Lipschitz. On the other hand, $f(x) = \sqrt{|x|}$ is not locally Lipscitz, as $f$ is not Lipschitz on any interval containing 0.

More generally, we will be interested in functions from $\mathbb{R}^n$ to $\mathbb{R}^m$. Lipschitzness for functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ is defined mutatis mutandis. In greatest generality, the defintion can be extended to maps between arbitrary metric spaces, but we will not deal with arbitrary metric spaces in this series of notes.

Exercise: Prove that every continuously differentiable function is locally Lipschitz.

Exercise: Prove that a continuously differentiable function with bounded derivative is globally Lipschitz.

Exercise: Prove that the composition of two Lipschitz functions is Lipschitz.

Exercise: Prove that the composition of two locally Lipcshitz functions is locally Lipschitz.