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.

These properties concern the domain, the codomain and the range of functions.

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:

Relative to negation:

Relative to a binary operation and an order:

Relative to topology and order:

Relative to measure and topology

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 ${\displaystyle \mapsto }$ symbol or Church's ${\displaystyle \lambda }$ is often used. Also, sometimes mathematicians notate a function's domain and codomain by writing e.g. ${\displaystyle f:A\rightarrow B}$. These notions extend directly to lambda calculus and type theory, respectively.

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