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 $$\{q_j\}_{j \isin \mathbb{N}}$$ be an enumeration of the rational numbers in the unit interval $[0, 1]$. Let $$\{a_j\}_{j \isin \N}$$ be positive real numbers with $$\sum_j a_j < \infty$$. Define $$g\colon [0, 1] \rarr \R$$ 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 $g′(x) := +∞$. 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 $$\{g(q_j)\}_{j \isin \mathbb{N}}.$$ 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 ${ f = 0} ∩ {g = 0 }$, 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 $E$.