# 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 ${\displaystyle {\sqrt[{3}]{x}}}$ denote the real cubic root of the real number ${\displaystyle x.}$ Let ${\displaystyle \{q_{j}\}_{j\in \mathbb {N} }}$ be an enumeration of the rational numbers in the unit interval ${\displaystyle [0,\,1].}$ Let ${\displaystyle \{a_{j}\}_{j\in \mathbb {N} }}$ be positive real numbers with ${\displaystyle \textstyle \sum _{j}a_{j}<\infty .}$ Define, for all ${\displaystyle x\in [0,\,1]}$

${\displaystyle g(x):=\sum _{j=0}^{\infty }\,a_{j}{\sqrt[{3}]{x-q_{j}}}.}$

Since for any ${\displaystyle x\in [0,\,1]}$ each term of the series is less than or equal to aj in absolute value, the series 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

${\displaystyle g^{\prime }(x):={\frac {1}{3}}\sum _{j=0}^{\infty }{\frac {a_{j}}{\sqrt[{3}]{(x-q_{j})^{2}}}}>0,}$

at any point where the sum is finite; also, at all other points, in particular, at any of the ${\displaystyle q_{j},}$ one has ${\displaystyle \textstyle g^{\prime }(x):=+\infty .}$ Since the image of ${\displaystyle g}$ is a closed bounded interval with left endpoint ${\displaystyle 0=g(0),}$ up to a multiplicative constant factor one can assume that g maps the interval ${\displaystyle [0,\,1]}$ onto itself. Since g is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its composition inverse ${\displaystyle f\,:=g^{-1}}$ has a finite derivative at any point, which vanishes at least in the points ${\displaystyle \{g(q_{j})\}_{j\in \mathbb {N} }.}$ These form a dense subset of ${\displaystyle [0,\,1]}$ (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 Gδ subset of the real line. By definition, for any Pompeiu function this set is a dense Gδ set, therefore by the Baire category theorem it is a residual set. In particular, it possesses uncountably many points.
• A linear combination af(x) + bg(x) of Pompeiu functions is a derivative, and vanishes on the set {f = 0} ∩ {g = 0}, which is a dense Gδ 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 Gδ 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 [ab] 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 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).