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 every 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 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 by
For each 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
at every point where the sum is finite; also, at all other points, in particular, at each of the, one has. Since the image of is a closed bounded interval with left endpoint
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 every point, which vanishes at least at the points These form a dense subset of .

Properties

It is known that the zero-set of a derivative of any everywhere differentiable 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.
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 .