List of types of functions

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

[edit] 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.

[edit] 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).

[edit] 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.

[edit] Relative to an ordering

• Monotonic function: does not reverse ordering of any pair.
• Strict Monotonic function: preserves the given order.

[edit] 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