Approximate differentiability according to Stepanoff-Whitney-Federer Chun-Liang LinFon-Che Liu 54C30Approximate limitApproximate limit superiorApproximate differentiabilityApproximate partial derivativeLipschitz continuityApproximate Lipschitz continuityLusin property 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.5024 10.1512/iumj.2013.62.5024 en Indiana Univ. Math. J. 62 (2013) 855 - 868 state-of-the-art mathematics http://iumj.org/access/