What does the Lipschitz constant tell us?

Question

Lipschitz continuity of a function is described in the following:

Let $f$ be a function where $f : \mathbb{R}^n \rightarrow \mathbb{R}$. $f$ is said to be Lipschitz continuous on $\mathbb{R}^n$ if there exists a constant $L>0$ such that, \begin{equation} \|f(\mathbf{y}) - f(\mathbf{x})\| \leq L\|\mathbf{y} - \mathbf{x}\|, \qquad \forall \mathbf{x},\mathbf{y} \in \mathbb{R}^n. \end{equation}

I am wondering, what does the Lipschitz constant tell us?

Answer 1

In the case $n= 1$, we can view Lipschitz continuity geometrically in the following way: for each point $a\in\mathbb R$, the graph is contained in the cone $$\{(x,y)\in\mathbb R^2: |y-f(a)|\leq L|x-a|\}$$ Thus Lipschitz continuity imposes that the function can't grow faster than linearly. This also imposes bounds on the derivatives of the function, wherever its differentiable, and Rademacher's theorem shows that this actually happens almost everywhere.