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.

Relative to set theory

These properties concern the domain, the codomain and the range 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, i.e. the codomain equals the range. 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.

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

• Continuous function: in which preimages of open sets 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, whose 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 power series.
• Arithmetic function: A function from the positive integers into the complex numbers.
• Differentiable function: Has a derivative.
• Smooth function: Has derivatives of all orders.
• Holomorphic function: Complex valued function of a complex variable which is differentiable at every point in its domain.
• Meromorphic function: Complex valued function that is holomorphic everywhere, apart from at isolated points where there are poles.
• Entire function: A holomorphic function whose domain is the entire 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 $\mapsto$ symbol or Church's $\lambda$ is often used. Also, sometimes mathematicians notate a function's domain and codomain by writing e.g. $f:A\rightarrow B$. 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 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.

References

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