Pompeiu Derivative

In mathematical analysis, a Pompeiu derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense set. In particular, 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 constructing an explicit example; these functions are therefore named after him.

Pompeiu's construction

Pompeiu's construction is described here. Let \sqrt /math> denote the real cube root of the real number . Let $\_$ be an enumeration of the rational numbers in the unit interval . Let $\_$ be positive real numbers with . Define $g:$, 1\rarr \mathbb by

:g(x): = a_0+\sum_^\infty \,a_j \sqrt

For any in , each term of the series is less than or equal to in absolute value, so the series uniformly converges to a continuous, strictly increasing function , by the Weierstrass -test. Moreover, it turns out that the function is differentiable, with

:$g^(x) := \frac\sum_^\infty \frac>0,$

at any point where the sum is finite; also, at all other points, in particular, at any of the , one has . Since the image of is a closed bounded interval with left endpoint

:g(0) = a_0-\sum_^\infty \,a_j \sqrt

up to the choice of , we can assume and up to the choice of a multiplicative factor we can assume that maps the interval onto itself. Since is strictly increasing it is injective, and hence a homeomorphism; and by the theorem of differentiation of the inverse function, its inverse has a finite derivative at any point, which vanishes at least at the points $\_.$ These form a dense subset of (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 (and more generally, of any Baire class one function) is a subset of the real line. By definition, for any Pompeiu function, this set is a ''dense'' set; therefore it is a residual set. In particular, it possesses uncountably many points.
* A linear combination of Pompeiu functions is a derivative, and vanishes on the set , which is a dense set by the Baire category theorem. Thus, Pompeiu functions form a vector space of functions.
* A limit function of a uniformly convergent sequence of Pompeiu derivatives is a Pompeiu 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 sequence: since these are dense sets, the zero set of the limit function is also dense.
* As a consequence, the class of all bounded Pompeiu derivatives on an interval 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 .

References

*
* Andrew M. Bruckner, "Differentiation of real functions"; CRM Monograph series, Montreal (1994).

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Real analysis