# Heuristic on Sobolev and BV functions

Let $$f: \Omega \subset \mathbb{R}^N \to \mathbb{R}^M$$ be a Sobolev or BV vector field.

A heuristic that I've heard frequently is the following:

$$f$$ is almost Lipschitz on a large "good" set but there is a small "bad" set where $$Df$$ is very large.

What theorems make this heuristic rigorous?

• In some sense, this follows from en.wikipedia.org/wiki/… applied to $Df$. – gerw Apr 15 at 18:01