What Lipschitz function can tell me about?

Question

If a function is said to be "Lipschitz", what kind of informations it can give? I know that is about continuity but, I think maybe it can give more informations about.

Answer 1

It is stronger than simple continuity. It is one of its strongest form. The greatest disadvantage I can see is that it is only defined on metric spaces which means that you cannot have a similar approach on general spaces.

But coming back to the real case. If you have a Lipschitz function $f:\mathbb{R}\rightarrow\mathbb{R}$, then it has many nice properties. The best one being differentiable almost everywhere. If you don't know that almost everywhere means, you can see it as follows (which is a stronger statement, but also true for every Lipschitz function): there is a bounded (locally integrable) function $g$ such that $$ f(x) = f(0) + \int_0^x g(y)\;dy. $$ In this case one usually writes $f' = g$. And the above says that $f'$ is bounded by some constant.