Approximate differentiability

2020 Mathematics Subject Classification: Primary: 26B05 Secondary: 28A2049Q15 [MSN][ZBL]

Definition

A generalization of the concept of differentiability obtained by replacing the ordinary limit by an approximate limit. Consider a (Lebesgure) measurable set $E\subset \mathbb R^n$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has Lebesgue density $1$. The map $f$ is approximately differentiable at $x_0$ if there is a linear map $A:\mathbb R^n\to \mathbb R^k$ such that \[ {\rm ap}\, \lim_{x\to x_0} \frac{f(x)-f(x_0) - A (x-x_0)}{|x-x_0|} = 0\, , \] (cp. with Section 6.1.3 of [EG] and Section 3.1.2 of [Fe]). $A$ is then called the approximate differential of $f$ at $x_0$. If $n=1$ (i.e. $E$ is a subset of the real line), the map $A$ takes the form $A (t) = a t$: the vector $a$ is then the approximate derivative of $f$ at $x_0$, and it is sometimes denoted by $f'_{ap} (x_0)$.

Properties

If $f$ is approximately differentiable at $x_0$, then it is approximately continuous at $x_0$. The usual rules about uniqueness of the differential, differentiability of sums, products and quotients of functions apply to approximate differentiable functions as well and follow from a useful characterization of approximate differentiability:

Proposition 1 Consider a (Lebesgue) measurable set $E\subset \mathbb R^n$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has Lebesgue density $1$. $f$ is approximately differentiable at $x_0$ if and only if there is a measurable set $F\subset E$ which has Lebesgue density $1$ at $x_0$ and such that $f|_F$ is classically differentiable at $x_0$. The approximate differential of $f$ at $x_0$ coincides then with the classical differential of $f|_F$ at $x_0$.

The chain rule applies to compositions $\varphi\circ f$ when $f$ is approximately differentiable at $x_0$ and $\varphi$ is classically differentiable at $f(x_0)$.

Stepanov and Federer's Theorems

The almost everywhere differentiabiliy of a function can be characterized in the following ways.

Theorem 2 (Stepanov) A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if and only if the approximate partial derivatives exist almost everywhere.

For the proof see Section 3.1.4 of [Fe]. A proof for the $2$-dimensional case can also be found in Section 12 of Chapter IX in [Sa]. According to [Sa] the notion of approximate differentiability in $2$ dimensions has been first introduced by Stepanov, who proved the $2$-dimensional case of Theorem 3. In the literature the name Stepanov theorem is usually attributed to another result in the differentiability of functions, see also Rademacher theorem.

Theorem 3 (Federer, Theorem 3.1.6 of [Fe]) Let $E\subset \mathbb R^n$ be a measurable set with finite measure. A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if for every $\varepsilon > 0$ there is a compact set $F\subset E$ such that $\lambda (E\setminus F)<\varepsilon$ and $f|_F$ is $C^1$ (i.e. there exists an extension $g$ of $f|_F$ to $\mathbb R^n$ which is $C^1$).

In the latter theorem it follows also that the classical differential of $f|_F$ coincides with the approximate differential of $f$ at almost every $x_0\in F$.

Notable examples of maps which are almost everywhere approximately differentiable are the ones belonging to the Sobolev classes $W^{1,p}$ and to the BV class (cp. with Theorem 4 of Section 6.1.3 of [EG]).

References

[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Br]	A.M. Bruckner, "Differentiation of real functions" , Springer (1978) MR0507448 Zbl 0382.26002
[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
[Mu]	M.E. Munroe, "Introduction to measure and integration" , Addison-Wesley (1953) MR035237 Zbl 0227.28001
[Sa]	S. Saks, "Theory of the integral" , Hafner (1952) MR0167578 Zbl 63.0183.05
[Th]	B.S. Thomson, "Real functions" , Springer (1985) MR0818744 Zbl 0581.26001