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{{MSC|28A33|49Q15}}
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{{MSC|26B05|28A20,49Q15}}
  
 
[[Category:Classical measure theory]]
 
[[Category:Classical measure theory]]
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{\rm ap}\, \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}
 
{\rm ap}\, \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}
 
\]
 
\]
exists at it is finite, then it is called approximate derivative of the function $f$ at $x_0$ (some authors include also the case in which the limit is $\pm infty$). In that case the function $f$ is called approximately differentiable at $x_0$ and, as a consequence of the definition, $f$ is [[Approximate continuity|approximately continuous]] at $x_0$. The concept can be extended further to functions of several variables: see [[Approximate differentiability]] Some authors denote the approximate derivative by $f'_{ap} (x_0)$, whereas some authors keep the notation $f' (x_0)$. Indeed if the classical derivative exists, then it coincides with the approximate derivative, whereas the opposite is false (note, for instance, that if $g$ coincides alomst everywhere with $f$, then $g$ is as well approximately differentiable at $x_0$).  
+
exists at it is finite, then it is called approximate derivative of the function $f$ at $x_0$ and the function is called approximately differentiable at $x_0$, see Section 3.1.2 of {{Cite|Fe}} and Section 67 of {{Cite|Th}} (when $k=1$ some authors include also the case in which the limit is $\pm\infty$). As a consequence of the definition, $f$ is [[Approximate continuity|approximately continuous]] at every $x_0$ where it is approximately differentiable. The concept can be extended further to functions of several variables: see [[Approximate differentiability]]. Some authors denote the approximate derivative by $f'_{ap} (x_0)$, whereas some authors keep the notation $f' (x_0)$. Indeed if the classical derivative exists, then it coincides with the approximate derivative, whereas the opposite is false (note, for instance, that if $g$ coincides almost everywhere with $f$, then $g$ is as well approximately differentiable at $x_0$).
  
 
====Properties====
 
====Properties====
The following useful proposition relates further the two concept.
+
The following useful proposition relates further the two concepts.
  
 
'''Proposition 1'''
 
'''Proposition 1'''
Let $E$, $f$ and $x_0$ be as above. The approximate derivative at $x_0$ exists if and only if there is a measurable set $F\subset E$ which has density $1$ at $x_0$ and such that the classical derivative exists for $f|_F$ at $x_0$. Moreover, the approximate derivative at $x_0$ equals the classical derivative of $f_F$ at the same point.
+
Let $E$, $f$ and $x_0$ be as above. The approximate derivative at $x_0$ exists if and only if there is a measurable set $F\subset E$ which has density $1$ at $x_0$ and such that the classical derivative exists for $f|_F$ at $x_0$. Moreover, the approximate derivative of $f$ at $x_0$ equals the classical derivative of $f|_F$ at the same point.
  
 
As a corollary of the previous proposition, the classical rules for the differentiation of a sum, a difference, a  product, and a quotient of functions apply to (finite) approximate  derivatives as well. The theorem on the differentiation of a compositition of approximately differentiable functions  does not apply in general. However it applies to $\varphi\circ f$ if $f$ is approximately differentiable at $x_0$ and $\varphi$ is '''classically differentiable''' at $f(x_0)$.
 
As a corollary of the previous proposition, the classical rules for the differentiation of a sum, a difference, a  product, and a quotient of functions apply to (finite) approximate  derivatives as well. The theorem on the differentiation of a compositition of approximately differentiable functions  does not apply in general. However it applies to $\varphi\circ f$ if $f$ is approximately differentiable at $x_0$ and $\varphi$ is '''classically differentiable''' at $f(x_0)$.
  
 
====Approximate Dini derivatives and Denjoy-Khinchin theorem====
 
====Approximate Dini derivatives and Denjoy-Khinchin theorem====
Approximate Dini derivatives are defined by analogy with ordinary Dini derivatives (cf. [[Dini derivative|Dini derivative]]): $D^+ (x_0)$, $D^- (x_0)$
+
Approximate Dini derivatives are defined by analogy with ordinary Dini derivatives (cf. [[Dini derivative|Dini derivative]]): $D^+ f (x_0)$, $D^- f (x_0)$
and $d^+ (x_0)$, $d^- (x_0)$ are, respectively, the right and left approximate upper limits and the right and left approximate lower limits of the quotient
+
and $d^+ f (x_0)$, $d^- f (x_0)$ are, respectively, the right and left approximate upper limits and the right and left approximate lower limits of the quotient
 
\[
 
\[
\frac{f(x)-f(x_0)}{x-x_0}
+
\frac{f(x)-f(x_0)}{x-x_0}\,
 
\]
 
\]
 +
(see Section 72 of {{Cite|Th}}). The following theorem applies.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a0128508.png" /> — lim sup from the right; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a0128509.png" /> — lim inf from the right; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a01285010.png" /> — lim sup from the left; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a01285011.png" /> — lim inf from the left; for example,
+
'''Theorem 2 (Denjoy-Khinchin)'''
 +
If $E\subset \R$ and $f:E\to \R$ are (Lebesgue) measurable, at least one of the following three alternatives holds at almost every point $x_0\in E$:  
 +
* either $f$ has a finite approximate derivative,
 +
* or $D^+ f (x_0)= D^- f (x_0)= \infty$,
 +
* or $d^+ f (x_0)= d^- f (x_0)=-\infty$.
  
<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/a012850/a01285012.png" /></td> </tr></table>
+
(Observe that the two last alternatives are not exclusive!). See Section 72 of {{Cite|Th}} for a proof: however in there the theorem is credited to Denjoy, Young and Saks.
  
The following Denjoy–Khinchin theorems apply. If a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a01285013.png" /> is finite and Lebesgue-measurable on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a01285014.png" />, then, at almost any point of this set, either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a01285015.png" /> has a finite approximate derivative, or
+
====Relation with the integral====
 +
As it happens for the ordinary derivative, however, the existence almost everywhere of the approximate derivative does not imply that the fundamental theorem
 +
of calculus applies. More precisely, there are examples of functions $f$ which are classically differentiable almost everywhere on the interval $[0,1]$ but such that the identity
 +
\[
 +
f (b)-f(a) =\int_a^b f' (t)\, dt
 +
\]
 +
fails on a set of positive measure of extrema $(a,b)\in [0,1]^2$ (cp. with the example in [[Absolute continuity]]).
  
<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/a012850/a01285016.png" /></td> </tr></table>
+
If, on the other hand, $F$ and $f$ are measurable functions on $[0,1]$ taking values in $\mathbb R^k$ and the identity
 +
\begin{equation}\label{e:fundamental}
 +
F(a)-F(0) =\int_0^a f (t)\, dt
 +
\end{equation}
 +
holds for a.e. $a\in [0,1]$, then the function $F$ can be redefined on a set of measure zero so that
 +
* $F$ is absolutely continuous
 +
* the identity \ref{e:fundamental} holds for ''every'' $a$
 +
* the function $F$ is almost everywhere differentiable and $F'=f$ almost everywhere.
  
If
+
====Approximate partial derivatives====
 
+
If $E\subset \mathbb R^n$ and $f: E\to\mathbb R^k$ are measurable, approximate partial derivatives can be defined as follows. Consider a system of coordinates
<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/a012850/a01285017.png" /></td> </tr></table>
+
$x_1, \dots , x_n$, a point $p=(p_1, \ldots , p_n)\in E$ and a coordinate $i$. If
 
+
* $E':=\{ h\in\mathbb R: (p_1, \ldots, p_{i-1}, h, p_{i+1}, \ldots , p_n)\in E\}$ is measurable,
is a Denjoy–Khinchin integral, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012850/a01285018.png" /> almost-everywhere in the interval under consideration (an ordinary derivative need not exist on a set of positive measure). This theorem explains the role played by approximate derivatives in the theory of integrals.
+
* the map $h\mapsto g(h):= f ((p_1, \ldots, p_{i-1}, h, p_{i+1}, \ldots , p_n)$ is measurable
 +
and the approximate limit
 +
\[
 +
L := {\rm ap} \lim_{h\to p_i} \frac{g(h)- g (p_i)}{h-p_i}
 +
\]
 +
exists and it is finite, then $L$ is the ''approximate partial derivative of $f$ at $p$ in the $i$-th direction'' (cp. with Section 3.1.2 of {{Cite|Fe}}).
  
 +
====Comments====
 
There exist continuous functions without an ordinary or an approximate derivative at any point of a given interval.
 
There exist continuous functions without an ordinary or an approximate derivative at any point of a given interval.
 
+
The concept of approximate derivative was introduced by A.Ya. Khinchin in 1916.
Approximate partial derivatives of functions of several real variables are considered as well.
 
 
 
====Comments====
 
The concept of an approximate derivative was introduced by >A.Ya. Khinchin in 1916.
 
 
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Saks,  "Theory of the integral" , Hafner (1952)  (Translated from French)</TD></TR></table>
+
{|
 
+
|-
 
+
|valign="top"|{{Ref|AFP}}||  L. Ambrosio, N.   Fusco, D.  Pallara, "Functions of bounded variations  and  free  discontinuity  problems". Oxford Mathematical Monographs. The    Clarendon  PressOxford University Press, New York, 2000.      {{MR|1857292}}{{ZBL|0957.49001}}
 
+
|-
  
For other references see [[Approximate limit|Approximate limit]].
+
|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|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan  (1968) {{MR|0151555}} {{ZBL|0197.03501}}
 +
|-
 +
|valign="top"|{{Ref|Ru}}||  W. Rudin, "Principles of  mathematical analysis", Third edition,  McGraw-Hill (1976) {{MR|038502}}  {{ZBL|0346.2600}} 
 +
|-
 +
|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 17:17, 18 August 2012

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

A generalization of the concept of a derivative, where the ordinary limit is replaced by an approximate limit.

Definition

Consider a (Lebesgure) measurable set $E\subset \mathbb R$, a measurable map $f:E\to \mathbb R^k$ and a point $x_0\in E$ where $E$ has Lebesgue density $1$. If the approximate limit \[ {\rm ap}\, \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0} \] exists at it is finite, then it is called approximate derivative of the function $f$ at $x_0$ and the function is called approximately differentiable at $x_0$, see Section 3.1.2 of [Fe] and Section 67 of [Th] (when $k=1$ some authors include also the case in which the limit is $\pm\infty$). As a consequence of the definition, $f$ is approximately continuous at every $x_0$ where it is approximately differentiable. The concept can be extended further to functions of several variables: see Approximate differentiability. Some authors denote the approximate derivative by $f'_{ap} (x_0)$, whereas some authors keep the notation $f' (x_0)$. Indeed if the classical derivative exists, then it coincides with the approximate derivative, whereas the opposite is false (note, for instance, that if $g$ coincides almost everywhere with $f$, then $g$ is as well approximately differentiable at $x_0$).

Properties

The following useful proposition relates further the two concepts.

Proposition 1 Let $E$, $f$ and $x_0$ be as above. The approximate derivative at $x_0$ exists if and only if there is a measurable set $F\subset E$ which has density $1$ at $x_0$ and such that the classical derivative exists for $f|_F$ at $x_0$. Moreover, the approximate derivative of $f$ at $x_0$ equals the classical derivative of $f|_F$ at the same point.

As a corollary of the previous proposition, the classical rules for the differentiation of a sum, a difference, a product, and a quotient of functions apply to (finite) approximate derivatives as well. The theorem on the differentiation of a compositition of approximately differentiable functions does not apply in general. However it applies to $\varphi\circ f$ if $f$ is approximately differentiable at $x_0$ and $\varphi$ is classically differentiable at $f(x_0)$.

Approximate Dini derivatives and Denjoy-Khinchin theorem

Approximate Dini derivatives are defined by analogy with ordinary Dini derivatives (cf. Dini derivative): $D^+ f (x_0)$, $D^- f (x_0)$ and $d^+ f (x_0)$, $d^- f (x_0)$ are, respectively, the right and left approximate upper limits and the right and left approximate lower limits of the quotient \[ \frac{f(x)-f(x_0)}{x-x_0}\, \] (see Section 72 of [Th]). The following theorem applies.

Theorem 2 (Denjoy-Khinchin) If $E\subset \R$ and $f:E\to \R$ are (Lebesgue) measurable, at least one of the following three alternatives holds at almost every point $x_0\in E$:

  • either $f$ has a finite approximate derivative,
  • or $D^+ f (x_0)= D^- f (x_0)= \infty$,
  • or $d^+ f (x_0)= d^- f (x_0)=-\infty$.

(Observe that the two last alternatives are not exclusive!). See Section 72 of [Th] for a proof: however in there the theorem is credited to Denjoy, Young and Saks.

Relation with the integral

As it happens for the ordinary derivative, however, the existence almost everywhere of the approximate derivative does not imply that the fundamental theorem of calculus applies. More precisely, there are examples of functions $f$ which are classically differentiable almost everywhere on the interval $[0,1]$ but such that the identity \[ f (b)-f(a) =\int_a^b f' (t)\, dt \] fails on a set of positive measure of extrema $(a,b)\in [0,1]^2$ (cp. with the example in Absolute continuity).

If, on the other hand, $F$ and $f$ are measurable functions on $[0,1]$ taking values in $\mathbb R^k$ and the identity \begin{equation}\label{e:fundamental} F(a)-F(0) =\int_0^a f (t)\, dt \end{equation} holds for a.e. $a\in [0,1]$, then the function $F$ can be redefined on a set of measure zero so that

  • $F$ is absolutely continuous
  • the identity \ref{e:fundamental} holds for every $a$
  • the function $F$ is almost everywhere differentiable and $F'=f$ almost everywhere.

Approximate partial derivatives

If $E\subset \mathbb R^n$ and $f: E\to\mathbb R^k$ are measurable, approximate partial derivatives can be defined as follows. Consider a system of coordinates $x_1, \dots , x_n$, a point $p=(p_1, \ldots , p_n)\in E$ and a coordinate $i$. If

  • $E':=\{ h\in\mathbb R: (p_1, \ldots, p_{i-1}, h, p_{i+1}, \ldots , p_n)\in E\}$ is measurable,
  • the map $h\mapsto g(h):= f ((p_1, \ldots, p_{i-1}, h, p_{i+1}, \ldots , p_n)$ is measurable

and the approximate limit \[ L := {\rm ap} \lim_{h\to p_i} \frac{g(h)- g (p_i)}{h-p_i} \] exists and it is finite, then $L$ is the approximate partial derivative of $f$ at $p$ in the $i$-th direction (cp. with Section 3.1.2 of [Fe]).

Comments

There exist continuous functions without an ordinary or an approximate derivative at any point of a given interval. The concept of approximate derivative was introduced by A.Ya. Khinchin in 1916.

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
[Ro] H.L. Royden, "Real analysis" , Macmillan (1968) MR0151555 Zbl 0197.03501
[Ru] W. Rudin, "Principles of mathematical analysis", Third edition, McGraw-Hill (1976) MR038502 Zbl 0346.2600
[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 derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximate_derivative&oldid=27401
This article was adapted from an original article by G.P. Tolstov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article