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 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 ${\displaystyle x\in [0,\,1]}$

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