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.
be an enumeration of the rational numbers in the unit interval [0, 1].
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
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).