Pompeiu derivative

Template:Short description 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 Template:Mvar. Let $\{q_j{\}}_{j\in \mathbb{N}}$ be an enumeration of the rational numbers in the unit interval Template:Math. Let $\{a_j{\}}_{j\in \mathbb{N}}$ be positive real numbers with $\sum _j{a}_j<\mathrm{\infty }$. Define $g:[0,1]\to ℝ$ by

$g(x):=a_0+\sum _{j=1}^{\mathrm{\infty }}{a}_j\sqrt[3]{x-q_j}.$

For each Template:Math in Template:Math, each term of the series is less than or equal to Template:Math in absolute value, so the series uniformly converges to a continuous, strictly increasing function Template:Math, by the [[Weierstrass M-test|Weierstrass Template:Mvar-test]]. Moreover, it turns out that the function Template:Mvar is differentiable, with

$g^{\prime }(x):=\frac{1}{3}\sum _{j=1}^{\mathrm{\infty }}\frac{a_j}{\sqrt[3]{(x-q_j{)}^2}}>0,$

at every point where the sum is finite; also, at all other points, in particular, at each of the Template:Math, one has Template:Math. Since the image of Template:Mvar is a closed bounded interval with left endpoint

$g(0)=a_0-\sum _{j=1}^{\mathrm{\infty }}{a}_j\sqrt[3]{q_j},$

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 Template:Mvar maps the interval Template:Math onto itself. Since Template:Mvar is strictly increasing it is injective, and hence a homeomorphism; and by the theorem of differentiation of the inverse function, its inverse Template:Math has a finite derivative at every point, which vanishes at least at the points $\{g(q_j){\}}_{j\in \mathbb{N}}.$ These form a dense subset of Template:Math (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 [[Gδ set|Template:Math subset]] of the real line. By definition, for any Pompeiu function, this set is a dense Template:Math set; therefore it is a residual set. In particular, it possesses uncountably many points.
• A linear combination Template:Math of Pompeiu functions is a derivative, and vanishes on the set Template:Math, which is a dense $G_{\delta }$ set by the Baire category theorem. Thus, Pompeiu functions form a vector space of functions.
• A limit function of a uniformly convergent sequence of Pompeiu derivatives is a Pompeiu 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 sequence: since these are dense Template:Math sets, the zero set of the limit function is also dense.
• As a consequence, the class Template:Math of all bounded Pompeiu derivatives on an interval Template:Math 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 Template:Math.

References

• Template:Cite journal
• Andrew M. Bruckner, "Differentiation of real functions"; CRM Monograph series, Montreal (1994).