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 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 (actually, it vanishes in many other points; see below).