Stepanov theorem

2020 Mathematics Subject Classification: Primary: 26B05 Secondary: 26B35 [MSN][ZBL]

A theorem proved by Stepanov about the differentiability of 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 [Fe]. Stepanov's theorem can be easily concluded from 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 [Ma]. The measurability assumption can be dropped.

References

[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe]	H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Ma]	J. Maly, A simple proof of the Stepanov theorem on differentiability almost everywhere. Exposition. Math. 17 (1999), no. 1, 59–61. MR1687460