Approximate derivative

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