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In [[mathematical analysis]], a '''Pompeiu derivative''' is a real-valued [[function (mathematics)|function]] of one real variable that is the [[derivative (mathematics)|derivative]] of an everywhere [[differentiable]] function and that vanishes in a [[dense set]]. Note in particular that a Pompeiu derivative is discontinuous at any point where it is not 0. Whether non-identically zero such functions may exist was a problem that arose in the context of early-1900s research on functional differentiability and integrability. The question was affirmatively answered by [[Dimitrie Pompeiu]] by constructiong an explicit example; these functions are therefore named after him.
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==Pompeiu's construction==
Pompeiu's construction is described here. Let <math>\sqrt[3]{x}</math> denote the real [[cubic root]] of the real number <math>x.</math> Let <math>\{q_j\}_{j\in \N}</math> be an enumeration of the rational numbers in the unit interval <math>[0,\,1].</math> Let <math>\{a_j\}_{j\in \N}</math> be positive real numbers with <math>\textstyle\sum_j a_j < \infty.</math> Define, for all <math>x\in [0,\,1]</math>
 
:<math>g(x):=\sum_{j=0}^\infty \,a_j \sqrt[3]{x-q_j}.</math>
 
Since for any <math>x\in[0,\,1]</math> each term of the series is less than or equal to ''a''<sub>j</sub> in absolute value,  the series [[uniform convergence|uniformly converges]] to a continuous, strictly increasing function g(x), due to the [[Weierstrass M-test]]. Moreover, it turns out that the function ''g'' is differentiable, with
:<math>g^{\prime}(x):=\frac{1}{3}\sum_{j=0}^\infty \frac{a_j}{\sqrt[3]{(x-q_j)^2}}>0,</math>
at any point where the sum is finite; also, at all other points, in particular,  at any of the <math>q_j,</math> one has <math>\textstyle g^{\prime}(x):=+\infty.</math> Since the [[image (function)|image]] of <math>g</math> is a closed bounded interval with left endpoint <math>0=g(0),</math> up to a multiplicative constant factor one can assume that ''g'' maps the interval <math>[0,\,1]</math> onto itself. Since ''g'' is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its [[inverse function|composition inverse]] <math>f\,:=g^{-1}</math> has a finite derivative at any point, which vanishes at least in the points <math>\{g(q_j)\}_{j\in \N}.</math> These form a dense subset of <math> [0,\,1]</math> (actually, it vanishes in many other points; see below).
 
==Properties==
* It is known that the zero-set of a derivative of any everywhere differentiable function is a [[G delta set|''G''<sub>δ</sub>]] subset of the real line. By definition, for any Pompeiu function this set is a ''dense'' G<sub>δ</sub> set, therefore by the [[Baire category theorem]] it is a  [[residual set]]. In particular, it possesses uncountably many points.
* A linear combination ''af''(''x'')&nbsp;+&nbsp;''bg''(''x'') of Pompeiu functions is a derivative, and vanishes on the set {''f''&nbsp;=&nbsp;0}&nbsp;∩&nbsp;{''g''&nbsp;=&nbsp;0}, which is a dense ''G''<sub>δ</sub> by the Baire category theorem. Thus, Pompeiu functions are a vector space of functions.
* A limit function of a uniformly convergent sequence of Pompeiu derivatives is a Pompeiou derivative. Indeed, it is a derivative, due to the theorem of limit under the sign of derivative. Moreover, it vanishes in the intersection of the zero sets of the functions of the sequences: since these are dense ''G''<sub>δ</sub> sets, the zero set of the limit function is also dense.
* As a consequence, the class ''E'' of all bounded Pompeiu derivatives on an interval [''a'',&nbsp;''b''] is a ''closed'' linear subspace of the Banach space of all bounded functions under the uniform distance (hence, it is a Banach space).
*Pompeiu's above construction of a ''positive'' function is a rather peculiar example of a Pompeiu's function: a theorem of Weil states that ''generically'' a Pompeiu derivative assumes both positive and negative values in dense sets, in the precise meaning that such functions constitute a residual set of the Banach space&nbsp;''E''.
 
==References==
* Pompeiu, Dimitrie, "Sur les fonctions dérivées"; Math. Ann. 63 (1907), no. 3, 326—332.
* Andrew M. Bruckner, "Differentiation of real functions"; CRM Monograph series, Montreal (1994).
 
{{DEFAULTSORT:Pompeiu Derivative}}
[[Category:Real analysis]]

Latest revision as of 02:42, 11 May 2014

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