Editing Singular function (section)

{{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''&nbsp;{{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]].