- 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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