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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. 
==Relative to [[set theory]]==
These properties concern the [[Domain (identity mathematics)|domain]], the [[codomain]] and the [[Range (mathematics)|range]] of functions.
* [[Injective function]]: has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function.
* [[Surjective function]]: has a [[preimage]] for every element of the [[codomain]], i.e. the codomain equals the range. Also called a surjection or [[onto function]].
* [[Bijective function]]: is both an [[injective function|injection]] and a [[surjection]], and thus [[Inverse function|invertible]].

* [[Identity function]]: maps any given element to itself.
* [[Constant function]]: has a fixed value regardless of arguments.
* [[Empty function]]: whose domain equals the [[empty set]].

==Relative to an operator (c.q. a [[group theory|group]] or other [[Mathematical structure|structure]])==
These properties concern how the function is affected by [[arithmetic]] operations on its operand.

The following are special examples of a [[homomorphism]] on a [[binary operation]]:
* [[Additive function]]: preserves the addition operation: ''f''(''x''+''y'') = ''f''(''x'')+''f''(''y'').
* [[Multiplicative function]]: preserves the multiplication operation: ''f''(''xy'') = ''f''(''x'')''f''(''y'').

Relative to [[negation]]:
* [[Even function]]: is symmetric with respect to the ''Y''-axis. Formally, for each ''x'': ''f''(''x'') = ''f''(&minus;''x'').
* [[Odd function]]: is symmetric with respect to the [[Origin (mathematics)|origin]]. Formally, for each ''x'': ''f''(&minus;''x'') = &minus;''f''(''x'').

Relative to a binary operation and an [[order theory|order]]:
* [[Subadditive function]]: 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]]: in which [[preimage]]s of [[open set]]s are open.
* [[Nowhere continuous]] function: is not continuous at any point of its domain (e.g. [[Dirichlet function]]).
* [[Homeomorphism]]: is an [[injective function]] that is also [[continuous function|continuous]], whose [[inverse function|inverse]] is continuous.

==Relative to an ordering==
* [[Monotonic function]]: does not reverse ordering of any pair.
* Strict [[Monotonic function]]: preserves the given order.

==Relative to the real/complex numbers==
* [[Analytic function]]: Can be defined locally by a [[Convergent series|convergent]] [[power series]].
* [[Arithmetic function]]: A function from the positive [[integers]] into the [[complex number]]s.
* [[Differentiable function]]: Has a [[derivative]].
* [[Smooth function]]: Has derivatives of all orders.
* [[Holomorphic function]]: [[Complex number|Complex]] valued function of a complex variable which is differentiable at every point in its domain.
* [[Meromorphic function]]: [[Complex number|Complex]] valued function that is holomorphic everywhere, apart from at isolated points where there are [[Pole (complex analysis)|poles]].
* [[Entire function]]: A [[holomorphic function]] whose domain is the entire [[complex number|complex plane]].

==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]]: is declared by "doing the reverse" of a given function (e.g. [[arcsine]] is the inverse of [[sine]]).
* [[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 <math>\mapsto</math> symbol or [[Alonzo Church|Church]]'s [[lambda calculus|<math>\lambda</math>]] is often used.  Also, sometimes mathematicians notate a function's [[domain of a function|domain]] and [[codomain]] by writing e.g. <math>f:A\rightarrow B</math>.  These notions extend directly to [[lambda calculus]] and [[type theory]], respectively.

==Relation to Category Theory==

[[Category Theory]] is a branch of mathematics that formalizes the notion of a special function via arrows or [[morphisms]].  A [[category (mathematics)|category]] 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 [[Function composition|composition]] is provided on morphisms, every object has one special morphism from it to itself called the [[identity (mathematics)|identity]] on that object, and composition and identities are required to obey certain relations.

In a so-called [[concrete category]], the objects are associated with mathematical structures like [[set (mathematics)|sets]], [[magmas]], [[group (mathematics)|groups]], [[ring (mathematics)|rings]], [[topological spaces]], [[vector spaces]], [[metric spaces]], [[order theory|partial orders]], [[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 [[Function (mathematics)|functions]], magma [[homomorphisms]], [[group homomorphisms]], ring homomorphisms, [[continuous functions]], [[linear transformations]] (or [[matrix (mathematics)|matrices]]), [[metric map]]s, [[monotonic function]]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]], [[basis (linear algebra)|basis]], finite [[Group representation|representation]], [[isomorphism]]) are definable purely in category theoretic terms (cf. [[monomorphism]], [[epimorphism]]).

Category theory has been suggested as a foundation for mathematics on par with [[set theory]] and [[type theory]] (cf. [[topos]]).

[[Allegory (category theory)|Allegory theory]]<ref>Peter Freyd, Andre Scedrov (1990). Categories, Allegories. Mathematical Library Vol 39. North-Holland. ISBN 978-0-444-70368-2.</ref> provides a generalization comparable to category theory for [[relation (mathematics)|relations]] instead of functions.

==References==
{{reflist}}

[[Category:Calculus]]
[[Category:Mathematics-related lists|Functions]]
[[Category:Number theory]]
[[Category:Types of functions| ]]
[[Category:Category theory]]