Pompeiu derivative explained

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 . Let \{q _{j\}} _{j} be an enumeration of the rational numbers in the unit interval . Let \{a _{j\}} _{j} be positive real numbers with \sum _{j} a _{j} < inf ty . Define g\colon [ 0 , 1 ] \rarr \R byg(x): = a _{0+\sum} ^{inf ty} ^{j =1} a _{j} \sqrt[ 3 ]{x - q _{j}.}

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

g'(x) : = (1)/(3) \sum ^{inf ty} _{j =1} (a _{j})/(\sqrt[ 3 ]{(x - q ^{2} _{j)}) }>0,

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

g( 0 ) = a _{0-\sum} ^{inf ty} ^{j =1} a _{j} \sqrt[ 3 ]{q _{j},}

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 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 \{g(q _{j)\}} _{j} . These form a dense subset of (actually, it vanishes in many other points; see below).

Properties

• It is known that the zero-set of a derivative of any everywhere differentiable function (and more generally, of any Baire class one function) is a subset of the real line. By definition, for any Pompeiu function, this set is a dense set; therefore it is a residual set. In particular, it possesses uncountably many points.
• A linear combination of Pompeiu functions is a derivative, and vanishes on the set