Functions

According to Eric Weisstein et al at Mathworld:

A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from A to B is an object f such that every $a\backslash \in A$ is uniquely associated with an object . A function is therefore a many-to-one (or sometimes one-to-one) relation. The set A of values at which a function is defined is called its domain, while the set B of values that the function can produce is called its range. The term "map" is synonymous with function.
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Unfortunately, the term "function" is also used to refer to relations that map single points in the domain to possibly multiple points in the range. These "functions" are called multivalued functions (or multiple-valued functions), and arise prominently in the theory of complex functions, where the presence of multiple values engenders the use of so-called branch cuts.