# Difference between revisions of "Singular function"

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2010 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].

How to Cite This Entry:
Singular function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_function&oldid=27862
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article