Functions, Theory of

a branch of mathematics concerned with general properties of functions. The two principal divisions of the theory of functions are the theory of functions of a real variable and the theory of functions of a complex variable.

In “classical” mathematical analysis, the fundamental objects of study are continuous functions that are defined on finite or infinite intervals and possess a certain amount of smoothness. In the second half of the 19th century, however, the development of mathematics made imperative the systematic investigation of more general types of functions, since it had been discovered that the limit of a sequence of continuous functions may be discontinuous. In other words, the class of continuous functions is not closed with respect to the most important operation of analysis—passage to the limit. Thus, functions defined by such classical means as trigonometric series are often discontinuous or nondif-ferentiable. The derivatives of continuous functions may be discontinuous for the same reason. Finally, differential equations arising in the study of physical problems may lack smooth solutions yet have solutions of a more general kind (provided that the concept of a solution is generalized in a suitable manner). It is important that an answer to the original physical problem is indeed provided by such generalized solutions (seeGENERALIZED FUNCTIONS). Circumstances such as the above gave rise to the development of the theory of functions of a real variable.

Individual facts of the theory of functions of a real variable were discovered in the 19th century—for example, the existence of nondifferentiable continuous functions, nonintegrable functions, and series of continuous functions with a discontinuous sum. These facts, however, were usually regarded as “exceptions to the rules” and were not included in a general framework. It was not until the 20th century, when set theory methods were placed at the foundation of the study of functions, that the modern theory of functions of a real variable developed in a systematic manner.

Three branches of the theory of functions of a real variable may be distinguished:

(1) Metric function theory, wherein properties of functions are studied by means of the measure of the sets where the functions obtain (seeMEASURE OF A SET). In metric function theory integration and differentiation are studied from a general point of view (seeINTEGRAL; DIFFERENTIAL; and DERIVATIVE). In addition, the concept of convergence of sequences of functions is generalized in various ways, and the structure of a wide variety of discontinuous functions is studied. The most important class of functions studied in metric function theory is the class of measurable functions.

(2) Descriptive function theory, whose principal object of study is the process of passing to the limit (seeBAIRE CLASSIFICATION).

(3) Constructive function theory, which investigates questions of representation of arbitrary functions by relevant analytic means (seeAPPROXIMATION AND INTERPOLATION OF FUNCTIONS).

The theory of functions of a complex variable is discussed in ANALYTIC FUNCTIONS.