Pompeiu derivative

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 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 constructing an explicit example; these functions are therefore named after him. 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 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 constructing an explicit example; these functions are therefore named after him. Pompeiu's construction is described here. Let x 3 {displaystyle {sqrt{x}}} denote the real cubic root of the real number x . {displaystyle x.} Let { q j } j ∈ N {displaystyle {q_{j}}_{jin mathbb {N} }} be an enumeration of the rational numbers in the unit interval [ 0 , 1 ] . {displaystyle .} Let { a j } j ∈ N {displaystyle {a_{j}}_{jin mathbb {N} }} be positive real numbers with ∑ j a j < ∞ . {displaystyle extstyle sum _{j}a_{j}<infty .} Define, for all x ∈ [ 0 , 1 ] {displaystyle xin } Since for any x ∈ [ 0 , 1 ] {displaystyle xin } 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 q j , {displaystyle q_{j},} one has g ′ ( x ) := + ∞ . {displaystyle extstyle g^{prime }(x):=+infty .} Since the image of g {displaystyle 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 [ 0 , 1 ] {displaystyle } onto itself. Since g is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its composition inverse f := g − 1 {displaystyle f,:=g^{-1}} has a finite derivative at any point, which vanishes at least in the points { g ( q j ) } j ∈ N . {displaystyle {g(q_{j})}_{jin mathbb {N} }.} These form a dense subset of [ 0 , 1 ] {displaystyle } (actually, it vanishes in many other points; see below).