In real 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[{3}]{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 [0, 1]. 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 [0,1]\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[{3}]{x-q_{j}}}.}
For each x in [0, 1], 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[{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 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[{3}]{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 [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 f := g−1 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 [0, 1] (actually, it vanishes in many other points; see below).
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