  We introduce a generalized version of the local Lipschitz number
$\textrm{lip}\,u$, and show that it can be used to characterize Sobolev
functions $u\in W_{\textrm{loc}}^{1,p}(\mathbb R^n)$, $1\le p\le \infty$, as
well as functions of bounded variation. This concept turns out to be fruitful
for studying, and for establishing new connections between, a wide range of
topics including fine differentiability, Rademacher's theorem, Federer's
characterization of sets of finite perimeter, regularity of maximal functions,
quasiconformal mappings, Alberti's rank one theorem, as well as generalizations
to metric measure spaces.
