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 be an enumeration of the rational numbers in the unit interval [0, 1] . Let be positive real numbers with. Define by For each x in [0, 1] , each term of the series is less than or equal to aj 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 at every point where the sum is finite; also, at all other points, in particular, at each of the qj , one has g′(x) := +∞ . Since the image of g is a closed bounded interval with left endpoint 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 These form a dense subset of [0, 1] (actually, it vanishes in many other points; see below).

Properties

Gδ subset]] of the real line. By definition, for any Pompeiu function, this set is a dense Gδ set; therefore it is a residual set. In particular, it possesses uncountably many points. 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. Gδ sets, the zero set of the limit function is also dense. 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). E .