List of types of functions

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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, the codomain and the image of functions.

  • Injective function: has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function. In other words, every element of the function's codomain is the image of at most one element of its domain.
  • Surjective function: has a preimage for every element of the codomain, that is, the codomain equals the image. Also called a surjection or onto function.
  • Bijective function: is both an injection and a surjection, and thus 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.
  • Set function: whose input is a set.
  • Choice function called also selector or uniformizing function: assigns to each set one of its elements.

Relative to an operator (c.q. a group or other 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(−x).
  • Odd function: is symmetric with respect to the origin. Formally, for each x: f(−x) = −f(x).

Relative to a binary operation and an 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[]

Relative to topology and order:

  • Semicontinuous function: upper or lower semicontinuous.
  • Right-continuous function: no jump when the limit point is approached from the right. Left-continuous function: similarly.
  • Locally bounded function: bounded around every point.

Relative to an ordering[]

Relative to the real/complex numbers[]

Relative to measurability[]

  • Measurable function: the preimage of each measurable set is measurable.
  • Borel function: the preimage of each Borel set is a Borel set.
  • Baire function called also Baire measurable function: obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions.
  • Singular function: continuous, with zero derivative almost everywhere, but non-constant.

Relative to measure[]

Relative to measure and topology

Ways of defining functions/relation to type theory[]

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's is often used. Also, sometimes mathematicians notate a function's domain and codomain by writing e.g. . These notions extend directly to lambda calculus and type theory, respectively.

Higher order functions[]

These are functions that operate on functions or produce other functions, see Higher order function. Examples are:

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 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 is provided on morphisms, every object has one special morphism from it to itself called the 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 sets, magmas, groups, rings, topological spaces, vector spaces, metric spaces, 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 functions, magma homomorphisms, group homomorphisms, ring homomorphisms, continuous functions, linear transformations (or matrices), metric maps, monotonic functions, 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, finite 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 theory[1] provides a generalization comparable to category theory for relations instead of functions.


Other functions[]

More general objects still called functions[]

See also[]

  • List of types of sets

References[]

  1. ^ Peter Freyd, Andre Scedrov (1990). Categories, Allegories. Mathematical Library Vol 39. North-Holland. ISBN 978-0-444-70368-2.
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