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A non-constant continuous [[Function of bounded variation|function of bounded variation]] whose derivative is zero almost-everywhere on the interval on which it is defined. Singular functions arise as summands in the [[Lebesgue decomposition|Lebesgue decomposition]] of functions of bounded variation. For example, every continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855501.png" /> of bounded variation on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855502.png" /> can be uniquely written in the form of a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855503.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855504.png" /> is an absolutely-continuous function (cf. [[Absolute continuity|Absolute continuity]]) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855506.png" /> is a singular function or identically zero.
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{{MSC|26A45}}
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855507.png" />. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855508.png" /> can be written in the form
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[[Category:Analysis]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s0855509.png" /></td> </tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555011.png" /> or 2 for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555012.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555014.png" /> is the [[Cantor set|Cantor set]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555015.png" /> or 2 for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555016.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555017.png" /> be the first subscript for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555018.png" />; if there are no such subscripts, then one takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555019.png" />. The function
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A terminology introduced by Lebesgue (see Section III in Chapter VIII of {{Cite|Le}}) to denote those non-constant continuous [[Function of bounded variation|functions of bounded variation]] $f:I\to \mathbb R$ such that $f'=0$ almost-everywhere on the interval $I$ of definition. Singular functions arise as summands in the [[Lebesgue decomposition|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 $g$
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with smallest total variation such that $g(a)=0$ and $f-g$ is absolutely continuous.  
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555020.png" /></td> </tr></table>
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De Giorgi and its school introduced instead the terminology "Cantor part" of $f$ (cp. with {{Cite|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 [[Generalized derivative|distributional derivative]] of $f$.
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For more on this topic we refer to [[Function of bounded variation]] and {{Cite|AFP}}.
  
is then a monotone singular function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555021.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Lebesgue"Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars  (1928)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,   "Theory of functions of a real variable" , '''1–2''' , F. Ungar  (1955–1961(Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.R. Halmos,  "Measure theory" , v. Nostrand  (1950)</TD></TR></table>
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{|
 
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|-
 
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|valign="top"|{{Ref|AFP}}||    L. Ambrosio, N.  Fusco, D.  Pallara, "Functions of bounded  variations  and  free  discontinuity  problems". Oxford Mathematical Monographs. The      Clarendon PressOxford University Press, New York, 2000.        {{MR|1857292}}{{ZBL|0957.49001}}
 
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|-
====Comments====
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|valign="top"|{{Ref|DS}}||      N. Dunford, J.T. Schwartz, "Linear operators. General theory",     '''1''', Interscience (1958{{MR|0117523}} {{ZBL|0635.47001}}
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555022.png" /> is well-defined (i.e. the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555023.png" /> does not depend on the chosen representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085550/s08555024.png" />); it is known as Lebesgue's singular function.
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|-
 
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|valign="top"|{{Ref|Ha}}|| P.R. Halmos,  "Measure theory", v. Nostrand  (1950) {{MR|0033869}} {{ZBL|0040.16802}}
====References====
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|-
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K.R. Stromberg,  "Introduction to classical real analysis" , Wadsworth (1981)</TD></TR></table>
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|valign="top"|{{Ref|Le}}|| H. Lebesgue,  "Leçons sur l'intégration et la récherche des fonctions  primitives", Gauthier-Villars (1928).
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|-
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|valign="top"|{{Ref|Na}}|| I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)
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|-
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|}

Revision as of 15:02, 9 September 2012

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. 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)
How to Cite This Entry:
Singular function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Singular_function&oldid=13838
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article