Pompeiu derivative
Encyclopedia
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
. Note in particular that a Pompeiu derivative is discontinuous at any 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 constructiong an explicit example; these functions are therefore named after him.
denote the real cubic root of the real number 
Let 
be an enumeration of the rational numbers in the unit interval 
Let 
be positive real numbers with 
Define, for all 
Since for any
each term of the series is less than or equal to aj in absolute value, the series uniformly converges to a continuous, strictly increasing function g(x), due to the Weierstrass M-test. Moreover, it turns out that the function g is differentiable, with
at any point where the sum is finite; also, at all other points, in particular, at any of the
one has 
Since the image of 
is a closed bounded interval with left endpoint 
up to a multiplicative constant factor one can assume that g maps the interval 
onto itself. Since g is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its composition inverse
has a finite derivative at any point, which vanishes at least in the points 
These form a dense subset of 
(actually, it vanishes in many other points; see below).
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, a Pompeiu derivative is a real-valued function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense set
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
. Note in particular that a Pompeiu derivative is discontinuous at any 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
Dimitrie Pompeiu
-Biography:After studying in Dorohoi and Bucharest, he went to France, where he studied mathematics at the University of Paris . He obtained a Ph.D. degree in mathematics in 1905 with a thesis, On the continuity of complex variable functions, written under the direction of Henri Poincaré...
by constructiong an explicit example; these functions are therefore named after him.
Pompeiu's construction
Pompeiu's construction is described here. Let denote the real cubic root of the real number Let be an enumeration of the rational numbers in the unit interval Let be positive real numbers with Define, for all Since for any
each term of the series is less than or equal to aj in absolute value, the series uniformly converges to a continuous, strictly increasing function g(x), due to the Weierstrass M-test. Moreover, it turns out that the function g is differentiable, withat any point where the sum is finite; also, at all other points, in particular, at any of the
one has Since the image of is a closed bounded interval with left endpoint up to a multiplicative constant factor one can assume that g maps the interval onto itself. Since g is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its composition inverseInverse function
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
has a finite derivative at any point, which vanishes at least in the points 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 is a Gδ subset of the real line. By definition, for any Pompeiu function this set is a dense Gδ set, therefore by the Baire category theoremBaire category theoremThe Baire category theorem is an important tool in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space....
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 Gδ by the Baire category theorem. Thus, Pompeiu functions are a vector space of functions.
- A limit function of a uniformly convergent sequence of Pompeiu derivatives is a Pompeiou 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 sequences: 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 [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.