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 x 3 denote the real cubic root of the real number x . Let { q j } j ∈ N be an enumeration of the rational numbers in the unit interval [ 0 , 1 ] . Let { a j } j ∈ N be positive real numbers with ∑ j a j < ∞ . Define, for all x ∈ [ 0 , 1 ]
g ( x ) := ∑ j = 0 ∞ a j x − q j 3 . Since for any x ∈ [ 0 , 1 ] each term of the series is less than or equal to aj in absolute value, the series uniformly converges to a continuous, strictly increasing function g(x), due to the Weierstrass M-test. Moreover, it turns out that the function g is differentiable, with
g ′ ( x ) := 1 3 ∑ j = 0 ∞ a j ( x − q j ) 2 3 > 0 , at any point where the sum is finite; also, at all other points, in particular, at any of the q j , one has g ′ ( x ) := + ∞ . Since the image of g is a closed bounded interval with left endpoint 0 = g ( 0 ) , up to a multiplicative constant factor one can assume that g maps the interval [ 0 , 1 ] onto itself. Since g is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its composition inverse f := g − 1 has a finite derivative at any point, which vanishes at least in the points { g ( q j ) } j ∈ N . These form a dense subset of [ 0 , 1 ] (actually, it vanishes in many other points; see below).
It is known that the zero-set of a derivative of any everywhere differentiable function is a Gδ subset of the real line. By definition, for any Pompeiu function this set is a dense Gδ set, therefore by the Baire category theorem 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 Gδ by the Baire category theorem. Thus, Pompeiu functions are a vector space of functions.A limit function of a uniformly convergent sequence of Pompeiu derivatives is a Pompeiou 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 sequences: since these are dense Gδ 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.