mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a

real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...

''f'' on the interval 'a'', ''b''is said to be singular if it has the following properties: *''f'' is

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...

on 'a'', ''b'' (**) *there exists a set ''N'' of measure 0 such that for all ''x'' outside of ''N,'' the

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

''f'' (''x'') exists and is zero; that is, the derivative of ''f'' vanishes

almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...

. *''f'' is non-constant on 'a'', ''b'' A standard example of a singular function is the

Cantor function In mathematics, the Cantor function is an example of a function (mathematics), function that is continuous function, continuous, but not absolute continuity, absolutely continuous. It is a notorious Pathological_(mathematics)#Pathological_exampl ...

, 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 In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamic ...

. 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 In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...

for a

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...

which is neither a

discrete random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers ...

(since the

probability Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an e ...

is zero for each point) nor an absolutely

continuous random variable In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spa ...

(since the

probability density In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values ...

is zero everywhere it exists). Singular functions occur, for instance, as sequences of spatially modulated phases or structures in

solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...

s and

magnet A magnet is a material or object that produces a magnetic field. This magnetic field is invisible but is responsible for the most notable property of a magnet: a force that pulls on other ferromagnetic materials, such as iron, steel, nickel, ...

s, described in a prototypical fashion by the

Frenkel–Kontorova model The Frenkel–Kontorova (FK) model is a fundamental model of low-dimensional nonlinear physics. The generalized FK model describes a chain of classical particles with nearest neighbor interactions and subjected to a periodic on-site substrate pote ...

and by the

ANNNI model In statistical physics, the axial (or anisotropic) next-nearest neighbor Ising model, usually known as the ANNNI model, is a variant of the Ising model. In the ANNNI model, competing ferromagnetic and antiferromagnetic exchange interactions couple ...

, as well as in some

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

s. Most famously, perhaps, they lie at the center of the

fractional quantum Hall effect The fractional quantum Hall effect (fractional QHE or FQHE) is the observation of precisely quantized plateaus in the Hall conductance of 2-dimensional (2D) electrons at fractional values of e^2/h, where ''e'' is the electron charge and ''h'' i ...

When referring to functions with a singularity

When discussing

mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...

in general, or more specifically

real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

or differential equations, it is common for a function which contains a

mathematical singularity In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For exa ...

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 distributional or

generalized function In mathematics, generalized functions are objects extending the notion of functions on real or complex numbers. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful for tr ...

analysis. A

weak derivative In mathematics, a weak derivative is a generalization of the concept of the derivative of a function (''strong derivative'') for functions not assumed differentiable, but only integrable, i.e., to lie in the L''p'' space L^1( ,b. The method o ...

is defined that allows singular functions to be used in

partial differential equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...

s, etc.

References

(**) This condition depends on the

references A reference is a relationship between Object (philosophy), objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. ...

* * * * {{DEFAULTSORT:Singular Function Fractal curves Types of functions

When referring to functions with a singularity

See also

References