- 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 asurjection , 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 theempty set .

*Inverse function : is declared by "doing the reverse" of a given function (e.g.arcsine is the inverse ofsine ).

*Injective function : has a distinct value for each distinct argument. Also called an injection or, sometimes, one-to-one function.

*Surjective function : has apreimage for every element of thecodomain , i.e. the codomain equals the range. Also called a surjection oronto 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.

*Additive function : preserves the addition operation: "f"("x"+"y") = "f"("x")+"f"("y").

*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 whichpreimage s ofopen set s are open.

*Nowhere continuous function: is not continuous at any point of its domain (e.g.Dirichlet function ).

*Homeomorphism : is aninjective function that is also continuous, whose inverse is continuous.**Relative to an ordering***

Monotonic function : does not reverse ordering of any pair.

* StrictMonotonic function : preserves the given order.**Relative to the real/complex numbers***

Analytic function : Can be defined locally by aconvergent power series .

*Arithmetic function : A function from the positiveintegers into thecomplex number s.

*Differentiable function : Has aderivative .

*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 : Aholomorphic function whose domain is the entire complex plane.other types of functions

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