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 ${\sqrt[{3}]{x}}$ denote the real cube root of the real number $x$ . Let $\{q_{j}\}_{j\in \mathbb {N} }$ be an enumeration of the rational numbers in the unit interval $[0, 1]$ . Let $\{a_{j}\}_{j\in \mathbb {N} }$ be positive real numbers with $\sum _{j}a_{j}<\infty$ . Define $g\colon [0,1]\rightarrow \mathbb {R}$ by

g(x):=a_{0} \sum _{j=1}^{\infty }\,a_{j}{\sqrt[{3}]{x-q_{j}}}.

For each $x$ in $[0, 1]$ , each term of the series is less than or equal to $a j$ in absolute value, so the series uniformly converges to a continuous, strictly increasing function $g (x)$ , by the Weierstrass $M$ -test. Moreover, it turns out that the function $g$ is differentiable, with

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

at every point where the sum is finite; also, at all other points, in particular, at each of the $q j$ , one has $g'(x) := \infty$ . Since the image of $g$ is a closed bounded interval with left endpoint

g(0)=a_{0}-\sum _{j=1}^{\infty }\,a_{j}{\sqrt[{3}]{q_{j}}},

up to the choice of $a_{0}$ , we can assume $g(0)=0$ and up to the choice of a multiplicative factor we can assume that $g$ maps the interval $[0, 1]$ onto itself. Since $g$ is strictly increasing it is injective, and hence a homeomorphism; and by the theorem of differentiation of the inverse function, its inverse $f := g -1$ has a finite derivative at every point, which vanishes at least at the points $\{g(q_{j})\}_{j\in \mathbb {N} }.$ These form a dense subset of $[0, 1]$ (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 (and more generally, of any Baire class one function) is a $G δ$ subset of the real line. By definition, for any Pompeiu function, this set is a dense $G δ$ set; therefore 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 ${f = 0} \cap {g = 0}$ , which is a dense $G_{\delta }$ 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 $G δ$ 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 $[a, b]$ 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 $E$ .

References

Pompeiu, Dimitrie (1907). "Sur les fonctions dérivées". Mathematische Annalen (in French). 63 (3): 326–332. doi:10.1007/BF01449201. MR 1511410.
Andrew M. Bruckner, "Differentiation of real functions"; CRM Monograph series, Montreal (1994).