# Function (mathematics)

functionfunctionsmathematical functionmathematical functionsempty functionfunctionalmapmappingfunctional relationshiprestriction
In mathematics, a function is a relation between sets that associates to every element of a first set exactly one element of the second set.wikipedia
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### History of the function concept

Historically
Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
The mathematical concept of a function emerged in the 17th century in connection with the development of the calculus; for example, the slope of a graph at a point was regarded as a function of the x-coordinate of the point.

### Argument of a function

argumentargumentsinput
(which is spoken aloud as f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).
In mathematics, an argument of a function is a value that must be provided to obtain the function's result.

### Graph of a function

graphgraphsgraphing
, called the graph of the function.
In mathematics, the graph of a function

### Binary relation

relationrelationsidentity relation
In mathematics, a function is a relation between sets that associates to every element of a first set exactly one element of the second set.
A function may be defined as a special kind of binary relation.

### Real-valued function

real-valuedrealnumerical function
For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable, and this phrase does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval; such a function is then called a partial function.
In mathematics, a real-valued function is a function whose values are real numbers.

### Variable (mathematics)

variablesvariableunknown
(which is spoken aloud as f of x), the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (one often says that f is a function of the variable x).
In addition to numbers, variables are commonly used to represent vectors, matrices and functions.

### Map (mathematics)

mappingmapmaps
Functions are also called maps or mappings, though some authors make some distinction between "maps" and "functions" (see section #Map).
In the sense of a function, a map is often associated with some sort of structure, particularly a set constituting the codomain.

### Mathematical analysis

analysisclassical analysisanalytic
Typically, this occurs in mathematical analysis, where "a function from X to Y " often refers to a function that may have a proper subset of X as domain.
These theories are usually studied in the context of real and complex numbers and functions.

### Partial function

total functionpartialtotal
For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable, and this phrase does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval; such a function is then called a partial function.
) is a function

### Function of a real variable

real functionreal variablefunctions of a real variable
For example, a "function from the reals to the reals" may refer to a real-valued function of a real variable, and this phrase does not mean that the domain of the function is the whole set of the real numbers, but only that the domain is a set of real numbers that contains a non-empty open interval; such a function is then called a partial function.
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers

### Codomain

valuesimagetarget
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.
In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall.

### Injective function

injectiveinjectionone-to-one
For example, a function is injective if the converse relation is univalent, where the converse relation is defined as Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.
In mathematics, an injective function or injection or one-to-one function is a function that maps distinct elements of its domain to distinct elements of its codomain.

### Choice function

selectionselection functionglobal choice function τ
(This point of view is used for example in the discussion of a choice function.)
A choice function (selector, selection) is a mathematical function f that is defined on some collection X of nonempty sets and assigns to each set S in that collection some element f(S) of S.

### Leonhard Euler

EulerLeonard EulerEuler, Leonhard
As first used by Leonhard Euler in 1734, functions are denoted by a symbol consisting generally of a single letter in italic font, most often the lower-case letters
He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.

### Ordered pair

ordered pairspairpairs
A function is uniquely represented by the set of all pairs
Cartesian products and binary relations (and hence functions) are defined in terms of ordered pairs.

### Serial relation

totaltotal relationserial
A total relation is a relation such that
On its domain, a function is serial.

### Range (mathematics)

rangerangestarget
The range of a function is the set of the images of all elements in the domain.
In mathematics, and more specifically in naïve set theory, the range of a function refers to the codomain of the function, though depending upon usage it can sometimes refer to the image.

### Lambda calculus

beta reductionλ-calculusuntyped lambda calculus
In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application.
Its namesake, the Greek letter lambda, is used in lambda expressions and lambda terms to denote binding a variable in a function.

### Dynamical system

dynamical systemsdynamic systemdynamic systems
In the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems.
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.

### Function application

applicationappliedwritten
In logic and the theory of computation, the function notation of lambda calculus is used to explicitly express the basic notions of function abstraction and application.
In mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range.

### Category theory

categorycategoricalcategories
In category theory and homological algebra, networks of functions are described in terms of how they and their compositions commute with each other using commutative diagrams that extend and generalize the arrow notation for functions described above.
Informally, category theory is a general theory of functions.

### Category (mathematics)

categorycategoriesobject
A simple example is the category of sets, whose objects are sets and whose arrows are functions.

### Parameter

parametersparametricargument
The index notation is also often used for distinguishing some variables called parameters from the "true variables".
Mathematical functions have one or more arguments that are designated in the definition by variables.

### Group homomorphism

homomorphismgroup homomorphismshomomorphisms
In particular map is often used in place of homomorphism for the sake of succinctness (e.g., linear map or map from G to H instead of group homomorphism from G to H).
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h : G → H such that for all u and v in G it holds that

### Continuous function

continuouscontinuitycontinuous map
Whichever definition of map is used, related terms like domain, codomain, injective, continuous have the same meaning as for a function.
In mathematics, a continuous function is a function that does not have any jumps or other discontinuities.