# List of types of functions

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List of types of functions

Functions can be classified according to the properties they have. These properties describe the functions behaviour under certain conditions.

Relative to set theory

These properties concern the domain, the codomain and the range of functions.
* Bijective function: is both an injective and a surjection, and thus invertible.
* Composite function: is formed by the composition of two functions "f" and "g", by mapping "x" to "f"("g"("x")).
* Constant function: has a fixed value regardless of arguments.
* Empty function: whose domain equals the empty set.
* Inverse function: is declared by "doing the reverse" of a given function (e.g. arcsine is the inverse of sine).
* 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.
* Identity function: maps any given element to itself.
* Piecewise function: is defined by different expressions at different intervals.

Relative to an operator (c.q. a group)

These properties concern how the function is affected by arithmetic operations on its operand.
* 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").
* 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.
* 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.

other types of functions

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