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Pompeiu derivative

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Concept in mathematical analysis

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 x 3 {\displaystyle {\sqrt{x}}} denote the real cube root of the real number x. Let { q j } j ∈ N {\displaystyle \{q_{j}\}_{j\in \mathbb {N} }} be an enumeration of the rational numbers in the unit interval . Let { a j } j ∈ N {\displaystyle \{a_{j}\}_{j\in \mathbb {N} }} be positive real numbers with ∑ j a j < ∞ {\displaystyle \sum _{j}a_{j}<\infty } . Define g : [ 0 , 1 ] → R {\displaystyle g\colon \rightarrow \mathbb {R} } by

g ( x ) := a 0 + ∑ j = 1 ∞ a j x − q j 3 . {\displaystyle g(x):=a_{0}+\sum _{j=1}^{\infty }\,a_{j}{\sqrt{x-q_{j}}}.}

For each x in , 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

g ′ ( x ) := 1 3 ∑ j = 1 ∞ a j ( x − q j ) 2 3 > 0 , {\displaystyle g'(x):={\frac {1}{3}}\sum _{j=1}^{\infty }{\frac {a_{j}}{\sqrt{(x-q_{j})^{2}}}}>0,}

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

g ( 0 ) = a 0 − ∑ j = 1 ∞ a j q j 3 , {\displaystyle g(0)=a_{0}-\sum _{j=1}^{\infty }\,a_{j}{\sqrt{q_{j}}},}

up to the choice of a 0 {\displaystyle a_{0}} , we can assume g ( 0 ) = 0 {\displaystyle g(0)=0} and up to the choice of a multiplicative factor we can assume that g maps the interval 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 has a finite derivative at every point, which vanishes at least at the points { g ( q j ) } j ∈ N . {\displaystyle \{g(q_{j})\}_{j\in \mathbb {N} }.} 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 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 { f = 0} ∩ {g = 0}, which is a dense G δ {\displaystyle 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 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 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

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