FUNCTION, in mathematics, a variable whose values are determined by those of one or more other variables. The word variable here denotes a symbol which stands for any one of a class of things called values of the variable. Although the terms thus defined apply to other entities as well as to numbers, the mathe matical theory of functions is in the main concerned with variables whose values are numbers. If y and x are two such variables, and if to each value of x corresponds one value of y and only one, then y is said to be a single-valued function of x. The functional relation is expressed by the equation y = f (x), in which the sym bol f may be replaced by some other letter. Here x is said to be the independent variable, and y the dependent variable. If more than one value of y corresponds to a value of x, the variable y is a many-valued function of x.
The set of values assigned to a variable is called the domain of that variable, and a theory of functions must rest on precise notions about domains. One of its goals should be a, classification of functions with respect to their properties in given domains. As with the natural sciences, however, a mere taxonomy based on the presence or absence of distinctive traits cannot be the sole objective. These traits, the processes connected with them, and their consequences are the matters of greatest interest. In the general subdivision of mathematics styled analysis (q.v.) the role of the theory of functions consists in a critical examination of the validity of analytical processes. It is concerned mainly with the idea of a limit, and with the consequences of that notion, upon which the infinitesimal calculus is based.
If the domains of both dependent and independent variables consist of real numbers only, the corresponding function is a real function of a real variable (or variables). If the independent variable has complex values (i.e., numbers of the form a+-bV —1 where a and b are real) we have a function of a complex variable. The theory of such functions is equivalent to that of pairs of real functions of two real variables. Historically, however, much of the theory of functions of a complex variable has been other wise developed, and its field has included topics analogous to those whose treatment for real variables is considered as lying in the province of the infinitesimal calculus, rather than in that of a general theory of functions.