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?