article

Title: Approximate differentiability according to Stepanoff-Whitney-Federer

Authors: Chun-Liang Lin and Fon-che Liu

Issue: Volume 62 (2013), Issue 3, 855-868

Abstract:

$A theorem of Stepanoff claims that approximate differentiability almost everywhere of a function $u$ is equivalent to existence almost everywhere of approximate partial derivatives of the function, while Whitney proved that approximate differentiability almost everywhere of $u$ is equivalent to the following Lusin type property: \begin{enumerate}[] \item(*) Given $\varepsilon>0$, there is a $C^1$ function $v$ on $\mathbb{R}^n$ such that \[ |\{x\in D:u(x)\neq v(x)\}|<\varepsilon. \] \end{enumerate} Federer then established that (*) is equivalent to having $u$ be approximately locally Lipschitz almost everywhere in the sense that \[ \mathrm{ap}\limsup_{y\to x}\frac{|u(y)-u(x)|}{|y-x|}<\infty \] holds almost everywhere. This paper extends these results to the case of approximate differentiability of general order $\gamma$ which is not necessarily an integer.$