# Functions that are approximately differentiable a.e

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.

• Right now we do: "P. Goldstein, P. Hajlasz, Appendix to ...". I suspect the other places (I also do not doubt that they exist) are titled similarly. If you couldn't find this reformulation in standard measure theory textbooks (I take it for granted that you checked Federer, Rudin, Bochkarev and such), most likely just nobody bothered to twist the standard definition this way in the main text. However, if I were writing a book and gave one definition, the other one would be assigned as an exercise. Are you sure that you haven't overlooked this possibility? – fedja Apr 5 '18 at 1:47
• In similar manner, can one define the approximate differentiability of a function $f:E\subset\mathbb R^n\to\mathbb R^m$? (It seems that if we take corresponding norms on both terms of the fraction in the above definition of approximate differentiability, then appriximate differentiability can be defined). Also, is it available in the literature? – Biswa Ranjan Datt Feb 19 '20 at 12:11

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.]

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 $$E_x:=\bigcup_{j=1}^\infty\bigcup_{k=k_j}^{k_{j+1}-1}E_x^{1/j}\cap (B_{2^{-k}}(x)\setminus B_{2^{-k-1}}(x)).$$