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- Singular_function abstract "In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties:f is continuous on [a, b]. (**)there exists a set N of measure 0 such that for all x outside of N the derivative f ′(x) exists and is zero, that is, the derivative of f vanishes almost everywhere.f is nondecreasing on [a, b].f(a) < f(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 is zero everywhere it exists).Singular functions occur, for instance, as sequences of spatially modulated phases or structures in solids and magnets, described in a prototypical fashion by the model of Frenkel and Kontorova and by the ANNNI model, as well as in some dynamical systems. Most famously, perhaps, they lie at the center of the fractional quantum Hall effect.".
- Singular_function thumbnail Devils-staircase.svg?width=300.
- Singular_function wikiPageID "51453".
- Singular_function wikiPageRevisionID "605915924".
- Singular_function hasPhotoCollection Singular_function.
- Singular_function subject Category:Fractal_curves.
- Singular_function subject Category:Types_of_functions.
- Singular_function type Abstraction100002137.
- Singular_function type Attribute100024264.
- Singular_function type Curve113867641.
- Singular_function type FractalCurves.
- Singular_function type Line113863771.
- Singular_function type Shape100027807.
- Singular_function comment "In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties:f is continuous on [a, b].".
- Singular_function label "Funkcja osobliwa".
- Singular_function label "Singular function".
- Singular_function label "Сингулярная функция".
- Singular_function sameAs Funkcja_osobliwa.
- Singular_function sameAs m.0dkt5.
- Singular_function sameAs Q2374166.
- Singular_function sameAs Q2374166.
- Singular_function sameAs Singular_function.
- Singular_function wasDerivedFrom Singular_function?oldid=605915924.
- Singular_function depiction Devils-staircase.svg.
- Singular_function isPrimaryTopicOf Singular_function.
