imported>ChaoticVermillion: Adding local short description: "Type of function", overriding Wikidata description "continuous monotonic nonconstant function with zero derivative almost everywhere"

2025-11-01T13:57:26Z

Adding local short description: "Type of function", overriding Wikidata description "continuous monotonic nonconstant function with zero derivative almost everywhere"

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{{Short description|Type of function}}
{{More citations needed|date=December 2009}}
[[Image:Devils-staircase.svg|thumb|right|450px|The graph of the winding number of the [[circle map]] is an example of a singular function.]]
In [[mathematics]], a [[real-valued function]] ''f'' on the [[interval (mathematics)|interval]] [''a'', ''b''] is said to be '''singular''' if it has the following properties:

*''f'' is [[continuous function|continuous]] on [''a'', ''b'']. (**)
*there exists a set ''N'' of [[measure (mathematics)|measure]] 0 such that for all ''x'' outside of ''N,'' the [[derivative]] ''f'' {{prime}}(''x'') exists and is zero; that is, the derivative of ''f'' vanishes [[almost everywhere]].
*''f'' is non-constant on [''a'', ''b''].

A standard example of a singular function is the [[Cantor function]], which is sometimes called the devil's staircase (a term also used for singular functions in general). There are, however, other functions that have been given that name. One is defined in terms of the [[circle map]].

If ''f''(''x'') = 0 for all ''x'' ≤ ''a'' and ''f''(''x'') = 1 for all ''x'' ≥ ''b'', then the function can be taken to represent a [[cumulative distribution function]] for a [[random variable]] which is neither a [[discrete random variable]] (since the [[probability]] is zero for each point) nor an absolutely [[continuous random variable]] (since the [[probability density function|probability density]] is zero everywhere it exists).

Singular functions occur, for instance, as sequences of spatially modulated phases or structures in [[solid]]s and [[magnet]]s, described in a prototypical fashion by the [[Frenkel–Kontorova model]] and by the [[ANNNI model]], as well as in some [[dynamical system]]s. Most famously, perhaps, they lie at the center of the [[fractional quantum Hall effect]].

==When referring to functions with a singularity==

When discussing [[mathematical analysis]] in general, or more specifically [[real analysis]] or [[complex analysis]] or [[differential equation]]s, it is common for a function which contains a [[mathematical singularity]] to be referred to as a 'singular function'. This is especially true when referring to functions which diverge to infinity at a point or on a boundary. For example, one might say, "''1/x'' becomes singular at the origin, so ''1/x'' is a singular function."

Advanced techniques for working with functions that contain singularities have been developed in the subject called [[Distribution (mathematics)|distributional]] or [[generalized function]] analysis. A [[weak derivative]] is defined that allows singular functions to be used in [[partial differential equation]]s, etc.

==See also==
* [[Absolute continuity]]
* [[Mathematical singularity]]
* [[Generalized function]]
* [[Distribution (mathematics)|Distribution]]
* [[Minkowski's question-mark function]]

==References==

(**) This condition depends on the [[references]]
<ref>{{SpringerEOM|title=Singular function}}</ref>
<references/>

* {{citation |last=Lebesgue |first=H. |title= Theory of functions of a real variable |year=1955–1961 |publisher=[[F. Ungar]] }}
* {{citation |last=Halmos |first=P.R. |title= Measure theory |year=1950 |publisher=[[v. Nostrand]] }}
* {{citation |last=Royden |first=H.L |title= Real Analysis |year=1988 |publisher=[[Prentice-Hall, Englewood Cliffs, New Jersey]] }}
* {{citation |last=Lebesgue |first=H. |title= Leçons sur l'intégration et la récherche des fonctions primitives |year=1928 |publisher=[[Gauthier-Villars]] }}

{{DEFAULTSORT:Singular Function}}
[[Category:Fractal curves]]
[[Category:Types of functions]]

Singular function - Revision history

imported>ChaoticVermillion: Adding local short description: "Type of function", overriding Wikidata description "continuous monotonic nonconstant function with zero derivative almost everywhere"