A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from to is an object such that every is uniquely associated with an object . A function is therefore a many-to-one (or sometimes one-to-one) relation. The set of values at which a function is defined is called its domain, while the set of values that the function can produce is called its range. Here, the set is called the codomain of .
In the context of univariate, real-valued functions , the fact that domain elements are mapped to unique range elements can be expressed graphically by way of the vertical line test.
In some literature, the term "map" is synonymous with function. Some caution must be exhibited, however, as it is not uncommon for the term map to denote a function with some sort of unspoken regularity assumption, e.g., in point-set topology, where "map" sometimes refers to a function which is continuous with respect to some topology.
Examples of functions over the reals include (many-to-one), (one-to-one), (two-to-one except for the single point ), etc.
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.
Several notations are commonly used to represent (non-multivalued) functions. The most rigorous notation is , which specifies that is function acting upon a single number (i.e., is a univariate, or one-variable, function) and returning a value . To be even more precise, a notation like ", where " is sometimes used to explicitly specify the domain and codomain of the function. The slightly different "maps to" notation is sometimes also used when the function is explicitly considered as a "map."
Generally speaking, the symbol refers to the function itself, while refers to the value taken by the function when evaluated at a point . However, especially in more introductory texts, the notation is commonly used to refer to the function itself (as opposed to the value of the function evaluated at ). In this context, the argument is considered to be a dummy variable whose presence indicates that the function takes a single argument (as opposed to , etc.). While this notation is deprecated by professional mathematicians, it is the more familiar one for most nonprofessionals. Therefore, unless indicated otherwise by context, the notation is taken in this work to be a shorthand for the more rigorous .