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- Types of functions

Integer-valued function

In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain. Floor and ceiling functions a

Test functions for optimization

In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as:
* Convergence rate.
* Precision.
* Robustness.
*

Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generaliz

Window function

In signal processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval, norm

Indicator function

In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset

Pseudoanalytic function

In mathematics, pseudoanalytic functions are functions introduced by Lipman Bers that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations.

Functional (mathematics)

In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author).
* In linear algebra, it is s

Coarse function

In mathematics, coarse functions are functions that may appear to be continuous at a distance, but in reality are not necessarily continuous. Although continuous functions are usually observed on a sm

Proto-value function

In applied mathematics, proto-value functions (PVFs) are automatically learned basis functions that are useful in approximating task-specific value functions, providing a compact representation of the

Barrier function

In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value on a point increases to infinity as the point approaches the boundary of the feasible regio

Quasiconvex function

In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form is a convex set.

Transcendental function

In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcen

Transfer function

In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a mathematical function that theoretically models the system's output f

Holonomic function

In mathematics, and more specifically in analysis, a holonomic function is a smooth function of several variables that is a solution of a system of linear homogeneous differential equations with polyn

Periodic function

A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of radians, are periodic functions. Periodic functio

Rvachev function

In mathematics, an R-function, or Rvachev function, is a real-valued function whose sign does not change if none of the signs of its arguments change; that is, its sign is determined solely by the sig

Concave function

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.

Bounded type (mathematics)

In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a fu

Ring of symmetric functions

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity. This ring

Liouvillian function

In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of

Even and odd functions

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical a

Support function

In mathematics, the support function hA of a non-empty closed convex set A in describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function

Constructible function

In complexity theory, a time-constructible function is a function f from natural numbers to natural numbers with the property that f(n) can be constructed from n by a Turing machine in the time of ord

Radially unbounded function

In mathematics, a radially unbounded function is a function for which Or equivalently, Such functions are applied in control theory and required in optimization for determination of compact spaces. No

Invex function

In vector calculus, an invex function is a differentiable function from to for which there exists a vector valued function such that for all x and u. Invex functions were introduced by Hanson as a gen

Global analytic function

In the mathematical field of complex analysis, a global analytic function is a generalization of the notion of an analytic function which allows for functions to have multiple branches. Global analyti

Zonal spherical function

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the mat

Radial function

In mathematics, a radial function is a function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. For example, a radial function

Radonifying function

In measure theory, a radonifying function (ultimately named after Johann Radon) between measurable spaces is one that takes a cylinder set measure (CSM) on the first space to a true measure on the sec

Almost periodic function

In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-pe

Subharmonic function

In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subha

Quasisymmetric function

In algebra and in particular in algebraic combinatorics, a quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring wit

Round function

In topology and in calculus, a round function is a scalar function , over a manifold , whose critical points form one or several connected components, each homeomorphic to the circle , also called cri

Coercive function

In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the contextdifferent exact definitions of this idea are in use.

Singular function

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

Progressive function

In mathematics, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform is supported by positive frequencies only: It is called super regressive if and only if the time reversed functio

Simple function

In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice

Unary function

A unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its range coincides with its domain. In contrast, a unary function's domain may

Positive-real function

Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis. They are complex functions, Z(s), of a complex v

Injective function

In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, f(x1) = f(x2) implies x

List of types of functions

Functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. A parabola is a specific type of function.

Transfer function matrix

In control system theory, and various branches of engineering, a transfer function matrix, or just transfer matrix is a generalisation of the transfer functions of single-input single-output (SISO) sy

Fabius function

In mathematics, the Fabius function is an example of an infinitely differentiable function that is nowhere analytic, found by Jaap Fabius. It was also written down as the Fourier transform of by Børge

Homogeneous function

In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called

Bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that for all x i

Morphism of algebraic varieties

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic va

Hermitian function

In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign: (where the indica

Continuous functions on a compact Hausdorff space

In mathematical analysis, and especially functional analysis, a fundamental role is played by the space of continuous functions on a compact Hausdorff space with values in the real or complex numbers.

Piecewise linear function

In mathematics and statistics, a piecewise linear, PL or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments.

Comparison function

In applied mathematics, comparison functions are several classes of continuous functions, which are used in stability theory to characterize the stability properties of control systems as Lyapunov sta

Jouanolou's trick

In algebraic geometry, Jouanolou's trick is a theorem that asserts, for an algebraic variety X, the existence of a surjection with affine fibers from an affine variety W to X. The variety W is therefo

Antiholomorphic function

In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of the complex variable

Weierstrass function

In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its di

Hardy–Littlewood maximal function

In mathematics, the Hardy–Littlewood maximal operator M is a significant non-linear operator used in real analysis and harmonic analysis. It takes a locally integrable function f : Rd → C and returns

Convex function

In mathematics, a real-valued function is called convex if the line segment between any two points on the graph of the function lies above the graph between the two points. Equivalently, a function is

Positive-definite function

In mathematics, a positive-definite function is, depending on the context, either of two types of function.

Hypertranscendental function

A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in Z

Baire one star function

A Baire one star function is a type of function studied in real analysis. A function is in class Baire* one, written , and is called a Baire one star function, if for each perfect set , there is an op

Pfaffian function

In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s

Identity function

In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. T

Slowly varying function

In real analysis, a branch of mathematics, a slowly varying function is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at

Kostant partition function

In representation theory, a branch of mathematics, the Kostant partition function, introduced by Bertram Kostant , of a root system is the number of ways one can represent a vector (weight) as a non-n

Surjective function

In mathematics, a surjective function (also known as surjection, or onto function) is a function f that every element y can be mapped from element x so that f(x) = y. In other words, every element of

Symmetrically continuous function

In mathematics, a function is symmetrically continuous at a point x if The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function is symme

List of window functions

In discrete-time signal processing, windowing is a preliminary signal shaping technique, usually applied to improve the appearance and usefulness of a subsequent Discrete Fourier Transform. Several wi

Measurable function

In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of an

Cardinal function

In mathematics, a cardinal function (or cardinal invariant) is a function that returns cardinal numbers.

Bijection

In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is pai

Representative function

No description available.

Bochner measurable function

In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable

Closed convex function

In mathematics, a function is said to be closed if for each , the sublevel setis a closed set. Equivalently, if the epigraph defined byis closed, then the function is closed. This definition is valid

Negligible function

In mathematics, a negligible function is a function such that for every positive integer c there exists an integer Nc such that for all x > Nc, Equivalently, we may also use the following definition.A

Direction-preserving function

In discrete mathematics, a direction-preserving function (or mapping) is a function on a discrete space, such as the integer grid, that (informally) does not change too drastically between two adjacen

Nowhere continuous function

In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f is a function from real numbers t

Polyconvex function

In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. Let Mm×n(K) denote the space of all m × n matrices over the fiel

Elementary function

In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rati

Vector-valued function

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. T

Supporting functional

In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.

Pseudoconvex function

In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but

Continuous function

In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This m

Binary function

In mathematics, a binary function (also called bivariate function, or function of two variables) is a function that takes two inputs. Precisely stated, a function is binary if there exists sets such t

Orthogonal functions

In mathematics, orthogonal functions belong to a function space that is a vector space equipped with a bilinear form. When the function space has an interval as the domain, the bilinear form may be th

Automorphic function

In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifol

Quasiperiodic function

In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function is quasiperiodic with quasiperiod if , where is a "simpler" function than . What

Weight function

A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The resu

Weakly measurable function

In mathematics—specifically, in functional analysis—a weakly measurable function taking values in a Banach space is a function whose composition with any element of the dual space is a measurable func

Proper convex function

In mathematical analysis, in particular the subfields of convex analysis and optimization, a proper convex function is an extended real-valued convex function with a non-empty domain, that never takes

Koenigs function

In mathematics, the Koenigs function is a function arising in complex analysis and dynamical systems. Introduced in 1884 by the French mathematician Gabriel Koenigs, it gives a canonical representatio

Fractal curve

A fractal curve is, loosely, a mathematical curve whose shape retains the same general pattern of irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal.

K-convex function

K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality of the policy in inventory control theory. The

Cauchy-continuous function

In mathematics, a Cauchy-continuous, or Cauchy-regular, function is a special kind of continuous function between metric spaces (or more general spaces). Cauchy-continuous functions have the useful pr

Regulated function

In mathematics, a regulated function, or ruled function, is a certain kind of well-behaved function of a single real variable. Regulated functions arise as a class of integrable functions, and have se

List of periodic functions

This is a list of some well-known periodic functions. The constant function f (x) = c, where c is independent of x, is periodic with any period, but lacks a fundamental period. A definition is given f

Algebraic function

In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms,

Baire function

In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced b

Locally integrable function

In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of d

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

Basis function

In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as e

Sublinear function

In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space is a real-valued function w

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