Functions that are approximately differentiable a.e

Question

The classical definition of an approximately differentiable function is as follows:

Definition. Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. We say that $f$ is approximately differentiable at $x\in E$ if there is a linear function $L:\mathbb{R}^n\to\mathbb{R}$ such that for any $\varepsilon>0$ the set $$ \left\{ y\in E:\, \frac{|f(y)-f(x)-L(y-x)|}{|y-x|} <\varepsilon \right\} \quad \text{has $x$ as a density point.} $$

It turns out that this definition is equivalent to the one described in the following result.

Theorem. A measurable function $f:E\to\mathbb{R}$ defined in a measurable set $E\subset\mathbb{R}^n$ is approximately differentiable at $x\in E$ if and only if there is a measurable set $E_x\subset E$ and a linear function $L:\mathbb{R}^n\to\mathbb{R}$ such that $x$ is a density point of $E_x$ and $$ \lim_{E_x\ni y\to x} \frac{|f(y)-f(x)-L(y-x)|}{|y-x|} = 0. $$

Do you know any reference for the proof of this result? Since we could not find a good reference, we proved it in the appendix of the paper listed below, but I am pretty sure it can be found somewhere else.

P. Goldstein, P. Hajłasz, A measure and orientation preserving homeomorphism with approximate Jacobian equal −1 almost everywhere. Arch. Ration. Mech. Anal. 225 (2017), 65–88.

In my opinion it is a folklore result that most of the people working with approximately differentiable functions know. The characterization given in the theorem is much easier to use than the original definition.

Answer 1

It turns out this is something I did some literature research on when I was working on my dissertation. Except for [6] (which I discovered too late for inclusion) and [7] (appeared later, but I happened to think of looking at it just now), the following from p. 35 of my dissertation are the pre-1993 references that I found. Later references probably exist, but this is not something I’ve kept track of over the years. Incidentally, [4] and [10] actually characterize, within the context of their generalized settings, when these two approaches agree.

[1] Arnaud Denjoy, Sur les fonctions dérivées sommables, Bulletin de la Société Mathématique de France 43 (1915), 161-248. [See pp. 167-168.]

[2] James Michael Foran, Fundamentals of Real Analysis, Monographs and Textbooks in Pure and Applied Mathematics #144, Marcel Dekker, 1991, xiv + 473 pages. [See pp. 272-275.]

[3] Donald Jay Geman and Joseph Horowitz, Occupation densities, Annals of Probability 8 #1 (February 1980), 1-67. [See pp. 23-24.]

[4] Hans Hahn and Arthur Rosenthal, Set Functions, The University of New Mexico Press, 1948, ix + 324 pages. [See p. 288.]

[5] Ernest William Hobson, The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Volume I, 3rd edition, Cambridge University Press, 1927, xvi + 736 pages (reprinted by Dover Publications in 1958). [See pp. 312-313.]

[6] Taqdir Husain, Topology and Maps, Mathematical Concepts and Methods in Science and Engineering #5, Plenum Publishing Company, 1977, xx + 337 pages. [See pp. 221-222.]

[7] Rangachary Kannan and Carole King Krueger, Advanced Analysis on the Real Line, Universitext, Springer-Verlag, 1996, x + 259 pages. [See pp. 43-44.]

[8] Fumitomo Maeda, On the definition and the approximate continuity of the general derivative, Journal of Science of the Hiroshima University (A) 2 (1932), 33-53 (abstract). [See p. 47.]

[9] Catherine V. Smallwood, Approximate upper and lower limits, Journal of Mathematical Analysis and Applications 37 #1 (January 1972), 223-227.

[10] Brian Sherif Thomson, Real Functions, Lecture Notes in Mathematics 1170, Springer-Verlag, 1985, viii + 229 pages. [See pp. 27-31.]

Answer 2

The equivalence between the two definitions is in fact really straightforward. Call $E_x^\varepsilon$ the set appearing in the first definition. Clearly the second definition implies the first one, as for $r$ small enough (depending on $\varepsilon$) $E_x\cap B_r(x)\subseteq E_x^\varepsilon$.

Conversely, let $k_0:=0$ and take, for each integer $j>0$, the smallest integer $k_j>k_{j-1}$ such that $$|E_x^{1/j}\cap(B_r(x)\setminus B_{r/2}(x))|\ge(1-1/j)|B_r(x)\setminus B_{r/2}(x)|$$ for all $r\le 2^{-k_j}$. Notice that you can pick $$E_x:=\bigcup_{j=1}^\infty\bigcup_{k=k_j}^{k_{j+1}-1}E_x^{1/j}\cap (B_{2^{-k}}(x)\setminus B_{2^{-k-1}}(x)).$$