REAL VARIABLE, Theory of Functions of the; a mathematical science. A variable is a symbol (such as x or y or "time) or 'temper ature))) which stands for any member of a class of things. A particular member of the class is called a "value) of the variable which is said to "take on) its values. One variable, y, is said to be a single-valued function of another variable, .r, if to every value of x there corre sponds one and only one value of y. This is the case, for example, if x stands for any num ber and y max'. It is customary to express such a relation between y and x by the equation r==f (s); x is called the independent and y the dependent variable. If the restriction that only owe value of y shall correspond to each value of x is removed, y is called a multiple-valued function of x. Unless otherwise provided, the word function in the following pages shall mean single-valued function.
While the notions of variable and function as thus defined apply to any class of objects whatever, they are usually thought of only in connection with classes of numbers. In fact the phrase Theory of Functions is at present understood by mathematicians to mean either the theory of functions of a complex variable (q.v.) or the theory of functions of a real vari able. In a certain abstract sense, the theory of the complex variable might be supposed to include the theory of the real variable, since all values of the real variable x, are included as special values (34=0) of the complex vari able xi As matter of fact, however, the theory of functions of a complex variable has yet concerned itself but little except with analytic functions i.e., those functions which may be thought of as defined by a power-series together with its continuations. This theory, therefore, leaves untouched the widest class of functions of the real variable. On the other hand, since the theory of functions of a com plex variable gives a broader and hence more unified theory of analytic functions than can be furnished by the real function theory, these functions are not ordinarily given special at tention in the latter theory. For a discussion
of algebraic functions the reader is referred to the article on Algebra; for transcendental func tions, i.e., functions which are not algebraic, see TatO0kroMETRY; HARMONIC ANALYSIS; EQUATIONS; SERIES, etc.
The theory of functions of a real variable is a product of the critical tendency of mathe matics in the latter half of the 19th century. It has yielded comparatively few new theorems about the functions hitherto studied in applied mathematics, and has therefore had less in fluence on these sciences than has the complex variable theory. Its main business has been the restatement in rigorous form and with proper limitation of generality, the brilliant though sometimes inexact theorems won by the earlier students of infinitesimal calculus. For example, the study of vibrating strings, membranes, etc., and of the conduction of heat led to the celebrated trigonometric series of J. B. J. Fourier (q.v.), which was at first in correctly supposed to furnish a method of repre senting an arbitrary function. The reconsid eration of this series and the attempt to deter mine exactly what functions are expansible by trigonometric series has led to many of the most abstruse theorems about point-sets. The theory of point-sets (Punktmengen,
The Real Variable.— The real numbers are, first, the whole numbers positive and negative; second, the rational fractions of the form min where m and n are whole numbers, and, third, the irrational numbers such as V2 and w. On this subject, see the articles on Aarrnmeric and ALGEBRA and the BIBLIOGRAPHY' at the end of this sketch. (Reference numbers such as that on the word Bibliography correspond to the numbers attached to works in the Bibliogra phy).