View source for Stepanov theorem

{{MSC|26B05|26B35}}

{{TEX|done}}

[[Category:Analysis]]

A theorem proved by Stepanov about the differentiability of [[Lipschitz Function|Lipschitz functions]] .

'''Theorem'''
Let $E\subset \mathbb R^m$ be measurable and $f: E \to \mathbb R^n$ a measurable function. Then
$f$ is a.e. differentiable on the set
\[
\left\{x\in E: \limsup_{y\to x} \frac{|f(x)-f(y)|}{|x-y|} < \infty \right\}\, .
\]

For a proof see Theorem 3.1.9 of {{Cite|Fe}}. Stepanov's theorem can be easily concluded from 
[[Rademacher theorem|Rademacher's theorem]]. This is classically done through Lebesgue's density theorem, cf. Theorem 1
in [[Density of a set]], but there is a an elementary derivation by Maly, see {{Cite|Ma}}. The measurability assumption can be  dropped.

====References====
{|
|-
|valign="top"|{{Ref|EG}}|| L.C. Evans, R.F. Gariepy, "Measure theory  and fine properties of functions" Studies in Advanced Mathematics. CRC  Press, Boca Raton, FL,  1992. {{MR|1158660}} {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Fe}}|| H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. {{MR|0257325}} {{ZBL|0874.49001}} 
|-
|valign="top"|{{Ref|Ma}}|| J. Maly, A simple proof of the Stepanov theorem on differentiability almost everywhere. ''Exposition. Math.'' '''17''' (1999), no. 1, 59–61. {{MR|1687460}}
|-
|}