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

Relative to set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

These properties concern the domain

Domain (mathematics)

In mathematics, the domain of definition or simply the domain of a function is the set of "input" or argument values for which the function is defined...

, the codomain

Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...

 and the range

Range (mathematics)

In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...

 of functions.

• Injective function
  Injective function
  In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
  
  : has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function.
• Surjective function
  Surjective function
  In mathematics, a function f from a set X to a set Y is surjective , or a surjection, if every element y in Y has a corresponding element x in X so that f = y...
  
  : has a preimage for every element of the codomain
  Codomain
  In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...
  
  , i.e. the codomain equals the range. Also called a surjection or onto function.
• Bijective function: is both an injection
  Injective function
  In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
  
   and a surjection, and thus invertible
  Inverse function
  In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
  
  .

• Identity function
  Identity function
  In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
  
  : maps any given element to itself.
• Constant function
  Constant function
  In mathematics, a constant function is a function whose values do not vary and thus are constant. For example the function f = 4 is constant since f maps any value to 4...
  
  : has a fixed value regardless of arguments.
• Empty function
  Empty function
  In mathematics, an empty function is a function whose domain is the empty set. For each set A, there is exactly one such empty functionf_A: \varnothing \rightarrow A....
  
  : whose domain equals the empty set
  Empty set
  In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
  
  .

Relative to an operator (c.q. a group

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

 or other structure

Mathematical structure

In mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....

)

These properties concern how the function is affected by arithmetic

Arithmetic

Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

 operations on its operand.

The following are special examples of a homomorphism

Homomorphism

In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

 on a binary operation

Binary operation

In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....

:

• Additive function
  Additive function
  In mathematics the term additive function has two different definitions, depending on the specific field of application.In algebra an additive function is a function that preserves the addition operation:for any two elements x and y in the domain. For example, any linear map is additive...
  
  : preserves the addition operation: f(x+y) = f(x)+f(y).
• Multiplicative function
  Multiplicative function
  In number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
  
  : preserves the multiplication operation: f(xy) = f(x)f(y).

Relative to negation

Negation

In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...

:

• Even function: is symmetric with respect to the Y-axis. Formally, for each x: f(x) = f(−x).
• Odd function: is symmetric with respect to the origin
  Origin (mathematics)
  In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...
  
  . Formally, for each x: f(−x) = −f(x).

Relative to a binary operation and an order

Order theory

Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

:

• Subadditive function
  Subadditive function
  In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive...
  
  : for which the value of f(x+y) is less than or equal to f(x)+f(y).
• Superadditive function: for which the value of f(x+y) is greater than or equal to f(x)+f(y).

Relative to a topology

• Continuous function
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
  : in which preimages of open set
  Open set
  The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
  
  s are open.
• Nowhere continuous
  Nowhere continuous
  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...
  
   function: is not continuous at any point of its domain (e.g. Dirichlet function).
• Homeomorphism
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
  : is an injective function
  Injective function
  In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
  
   that is also continuous
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
  , whose inverse
  Inverse function
  In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
  
   is continuous.

Relative to an ordering

• Monotonic function
  Monotonic function
  In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
  
  : does not reverse ordering of any pair.
• Strict Monotonic function
  Monotonic function
  In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....
  
  : preserves the given order.

Relative to the real/complex numbers

• Analytic function
  Analytic function
  In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
  
  : Can be defined locally by a convergent power series.
• Arithmetic function
  Arithmetic function
  In number theory, an arithmetic function is a real or complex valued function ƒ defined on the set of natural numbers In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers In number theory, an arithmetic (or...
  
  : A function from the positive integers into the complex number
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
  s.
• Differentiable function
  Differentiable function
  In calculus , a differentiable function is a function whose derivative exists at each point in its domain. The graph of a differentiable function must have a non-vertical tangent line at each point in its domain...
  
  : Has a derivative
  Derivative
  In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
  
  .
• Smooth function
  Smooth function
  In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
  
  : Has derivatives of all orders.
• Holomorphic function
  Holomorphic function
  In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
  
  : Complex
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
   valued function of a complex variable which is differentiable at every point in its domain.
• Meromorphic function
  Meromorphic function
  In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
  
  : Complex
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
   valued function that is holomorphic everywhere, apart from at isolated points where there are poles.
• Entire function
  Entire function
  In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
  
  : A holomorphic function
  Holomorphic function
  In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain...
  
   whose domain is the entire complex plane
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
  .

Ways of defining functions/Relation to Type Theory

• Composite function: is formed by the composition of two functions f and g, by mapping x to f(g(x)).
• Inverse function
  Inverse function
  In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
  
  : is declared by "doing the reverse" of a given function (e.g. arcsine is the inverse of sine
  Sine
  In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
  
  ).
• Piecewise function: is defined by different expressions at different intervals.

In general, functions are often defined by specifying the name of a dependent variable, and a way of calculating what it should map to. For this purpose, the symbol or Church

Alonzo Church

Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...

's

Lambda calculus

In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...

 is often used. Also, sometimes mathematicians notate a function's domain

Domain

-General:*Territory , a non-sovereign geographic area which has come under the authority of another government*Public domain, a body of works and knowledge without proprietary interest...

 and codomain

Codomain

In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...

 by writing e.g. . These notions extend directly to lambda calculus

Lambda calculus

In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...

 and type theory

Type theory

In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...

, respectively.

Relation to Category Theory

Category Theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

 is a branch of mathematics that formalizes the notion of a special function via arrows or morphisms. A category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 is an algebraic object that (abstractly) consists of a class of objects, and for every pair of objects, a set of morphisms. A partial (equiv. dependently typed) binary operation called composition

Composition

Composition may refer to:* Composition , in which one assumes that a whole has a property solely because its various parts have that property* Compounding is also known as composition in linguistic literature* in computer science...

 is provided on morphisms, every object has one special morphism from it to itself called the identity

Identity (mathematics)

In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...

 on that object, and composition and identities are required to obey certain relations.

In a so-called concrete category

Concrete category

In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions...

, the objects are associated with mathematical structures like sets, magmas, groups

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, rings

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

, topological spaces, vector spaces, metric spaces, partial orders

Order theory

Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

, differentiable manifolds, uniform spaces, etc., and morphisms between two objects are associated with structure-preserving functions between them. In the examples above, these would be functions

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

, magma homomorphisms, group homomorphisms, ring homomorphisms, continuous functions, linear transformations (or matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

), metric maps, monotonic function

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

s, differentiable functions, and uniformly continuous functions, respectively.

As an algebraic theory, one of the advantages of category theory is to enable one to prove many general results with a minimum of assumptions. Many common notions from mathematics (e.g. surjective, injective, free object

Free object

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....

, basis

Basis

Basis may refer to* Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation.* Basis of futures, the value differential between a future and the spot price...

, finite representation

Representation

Representation can refer to:* Representation , one's ability to influence the political process* Representative democracy* Representation, a type of diplomatic mission...

, isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

) are definable purely in category theoretic terms (cf. monomorphism

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

, epimorphism

Epimorphism

In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

).

Category theory has been suggested as a foundation for mathematics on par with set theory

Set theory

Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

 and type theory

Type theory

In mathematics, logic and computer science, type theory is any of several formal systems that can serve as alternatives to naive set theory, or the study of such formalisms in general...

 (cf. topos

Topos

In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...

).

Allegory theory

Allegory (category theory)

In mathematical category theory, an allegory is a category that has some of the structure of the category of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation...

 provides a generalization comparable to category theory for relations

Relation (mathematics)

In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...

instead of functions.