Pompeiu derivative

Pompeiu's construction

Pompeiu's construction is described here. Let \sqrt[3]{x} denote the real cube root of the real number x. Let {q_j}{j \isin \mathbb{N}} be an enumeration of the rational numbers in the unit interval [0, 1]. Let {a_j}{j \isin \N} be positive real numbers with \sum_j a_j . Define g\colon [0, 1] \rarr \R by :g(x): = a_0+\sum_{j=1}^\infty ,a_j \sqrt[3]{x-q_j}.

For each x in [0, 1], each term of the series is less than or equal to a**j 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 :g'(x) := \frac{1}{3} \sum_{j=1}^\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 q**j, one has . Since the image of g is a closed bounded interval with left endpoint :g(0) = a_0-\sum_{j=1}^\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 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 has a finite derivative at every point, which vanishes at least at the points {g(q_j)}_{j \isin \mathbb{N}}. These form a dense subset of [0, 1] (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 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.
• A linear combination af(x) + bg(x) of Pompeiu functions is a derivative, and vanishes on the set }, 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 G**δ 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.

References

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