Singular function

2020 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL]

A terminology introduced by Lebesgue (see Section III in Chapter VIII of [Le]) to denote those non-constant continuous functions of bounded variation $f:I\to \mathbb R$ such that $f'=0$ almost-everywhere on the interval $I$ of definition. These functions can also be characterized as those continuous functions whose distribution derivatives are singular measures without atoms. Singular functions arise as summands in the Lebesgue decomposition of functions of bounded variation. In particular, every continuous function $f:[a,b]\to\mathbb R$ of bounded variation can be written in a unique way (up to constants) as a sum of an absolutely continuous function $f_a$ and a singular function $f_c$. Lebesgue calls $f_c$ "fonction des singularités" of $f$ and defines it as that continuous function $g$ with smallest total variation such that $g(a)=0$ and $f-g$ is absolutely continuous.

De Giorgi and its school introduced instead the terminology "Cantor part" of $f$ (cp. with [AFP]). This name is justified by the fact that the Cantor ternary function is the best known example of singular function. The concept of Cantor part has been generalized by De Giorgi and its school to functions of bounded variations $f$ of several variables: however in this case the object which is decomposed is the distributional derivative of $f$. For more on this topic we refer to Function of bounded variation and [AFP].

References

[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958) MR0117523 Zbl 0635.47001
[Ha]	P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Le]	H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars (1928).
[Na]	I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)