Lipschitz continuous explained through example My brain can't grasp the concept of Lipschitz continuity for some how...
The definition is stated here: Lipschitz continuous
Now, can someone please run through an example and explain how we can determine whether a function is lipschitz continuous?
Let's look at the function $f(x,y)=x^2-x^3$ for instance... (or any other simple function). Is that  lipschitz continuous, and why?
 A: Common examples of Lipschitz functions are continuously differentiable functions with certain domains. If $f \in C^1([a, b], \mathbb{R})$, then $f$ is Lipschitz continuous since for $x, y \in [a, b]$ by the mean value theorem,
$$|f(x) - f(y)| \leq \sup_{[a, b]}|f'| \cdot |x - y|.$$
$\sup_{[a, b]}|f'|$ exists since $f'$ is continuous on the compact set $[a, b]$. Also if $\Omega \subset \mathbb{R}^n$, $f \in C^1(\Omega, \mathbb{R}^m)$, and $K \subset \Omega$ is compact and convex (actually the convexity of $K$ is not necessary), then $f|_{K}$ is Lipschitz (a proof is given here by zhw $C^1$ function on compact set is Lipschitz). So these are some commonly encountered Lipschitz functions.
A: To address your request, let's try to determine whether $f:\mathbb{R}^2\to\mathbb{R}$ given by $f(x,y)=x^2-x^3$ is Lipschitz or not. I often find that proving a function is not Lipschitz is harder than proving that it is Lipschitz.
I claim that $f$ is not Lipschitz. Indeed let $K\geq 0$. We need to produce $(x,y)$ and $(z,w)$ in $\mathbb{R}^2$ such that $$|f(x,y)-f(z,w)|>K\sqrt{(x-y)^2+(z-w)^2} $$
Choose $(z,w)=(0,0)$ and $(x,y)=(\sqrt{K+1}+1,0)$. Notice that,
$$|x||x-1|-K=(\sqrt{K+1}+1)(\sqrt{K+1})-K=1+\sqrt{K+1}>0$$
which implies
$$|x^3-x^2|-K|x|>0$$ and finally $$|x^3-x^2|>K|x|$$
hence $f$ is not Lipschitz.
If anything is unclear, please ask in the comments.
