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A generalization of the concept of differentiability obtained by replacing the ordinary limit by an [[Approximate limit|approximate limit]]. A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128601.png" /> of a real variable is called approximately differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128602.png" /> if there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128603.png" /> such that
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{{MSC|26B05|28A20,49Q15}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128604.png" /></td> </tr></table>
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[[Category:Classical measure theory]]
  
The magnitude <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128605.png" /> is called the approximate differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128606.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128607.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128608.png" /> is approximately differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a0128609.png" /> if and only if it has an [[Approximate derivative|approximate derivative]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286010.png" /> at this point. Approximate differentiability of real functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286011.png" /> real variables is defined in a similar manner. For example, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286013.png" /> is called approximately differentiable at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286014.png" /> if
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286015.png" /></td> </tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286017.png" /> are certain given numbers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286018.png" />. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286019.png" /> is called the approximate differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286020.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286021.png" />.
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====Definition====
 +
A generalization of the concept of differentiability obtained by replacing the ordinary limit by an [[Approximate limit|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 [[Density of a set|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 {{Cite|EG}} and Section 3.1.2 of {{Cite|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|approximate derivative]] of $f$ at $x_0$, and it is sometimes denoted by $f'_{ap} (x_0)$.
  
Stepanov's theorem: A real-valued measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286022.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286023.png" /> is approximately differentiable almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286024.png" /> if and only if it has finite approximate partial derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286025.png" /> and to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286026.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286027.png" />; these partial derivatives almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286028.png" /> coincide with the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012860/a01286030.png" />, respectively, of the approximate differential.
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====Properties====
 +
If $f$ is approximately differentiable at $x_0$, then it is [[Approximate continuity|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:
  
The concept of approximate differentiability can also be extended to vector functions of one or more real variables.
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'''Proposition 1'''
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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 $x_0$.
  
====References====
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The chain rule applies to compositions $\varphi\circ f$ when $f$ is approximately differentiable at $x_0$ and $\varphi$ is '''classically differentiable''' at $f(x_0)$.
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1952) (Translated from French)</TD></TR></table>
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 +
====Stepanov and Federer's Theorems====
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The almost everywhere differentiabiliy of a function can be characterized in the following ways.
 +
 
 +
'''Theorem 2 (Stepanov)'''
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A function $f:E\to\mathbb R^k$ is approximately differentiable almost everywhere if and only if the [[Approximate derivative|approximate partial derivatives]] exist almost everywhere.
 +
 
 +
For the proof see Section 3.1.4 of {{Cite|Fe}}. A proof for the $2$-dimensional case can also be found in Section 12 of Chapter IX in {{Cite|Sa}}. According to {{Cite|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 {{Cite|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 (of functions)|Sobolev classes]] $W^{1,p}$ and to the [[Function of bounded variation|BV class]] (cp. with Theorem 4 of Section 6.1.3 of {{Cite|EG}}).
  
====Comments====
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====References====
For other references see [[Approximate limit|Approximate limit]].
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{|
 +
|-
 +
|valign="top"|{{Ref|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.    {{MR|1857292}}{{ZBL|0957.49001}}
 +
|-
 +
|valign="top"|{{Ref|Br}}|| A.M. Bruckner,  "Differentiation of real  functions" , Springer  (1978) {{MR|0507448}}  {{ZBL|0382.26002}}
 +
|-
 +
|valign="top"|{{Ref|EG}}||  L.C. Evans, R.F. Gariepy,  "Measure theory  and fine properties of  functions" Studies in Advanced  Mathematics. CRC  Press, Boca Raton, FL,  1992. {{MR|1158660}}  {{ZBL|0804.2800}}
 +
|-
 +
|valign="top"|{{Ref|Fe}}||    H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren    der mathematischen Wissenschaften. Springer-Verlag New York Inc., New    York, 1969. {{MR|0257325}} {{ZBL|0874.49001}}
 +
|-
 +
|valign="top"|{{Ref|Mu}}||  M.E. Munroe,  "Introduction to measure and integration" ,  Addison-Wesley  (1953) {{MR|035237}} {{ZBL|0227.28001}}
 +
|-
 +
|valign="top"|{{Ref|Sa}}|| S. Saks,  "Theory of the integral" , Hafner  (1952) {{MR|0167578}} {{ZBL|63.0183.05}}
 +
|-
 +
|valign="top"|{{Ref|Th}}|| B.S. Thomson,  "Real functions" , Springer  (1985) {{MR|0818744}} {{ZBL|0581.26001}}
 +
|}

Latest revision as of 11:57, 2 May 2014

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
How to Cite This Entry:
Approximate differentiability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_differentiability&oldid=13557
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article