Another outstanding problem in connection with continuous functions is that of the orders of infinites and infinitesimals. A function f(x) becomes infinite or infinitesimal at a point if it approaches + co, — cc, or 0. If f(x) and # (x) both become infinite (or infinitesimal) at the same point, a, we may have their quotient ap proachin4 (1) more than one limit, (2) a unique finite limit, not zero, (3) zero, (4) c4. For the first case no extensive theory has been de veloped. In the second case the functions are said to be of the same order; in the third f(x) has an infinity of lower order or is infini tesimal of higher order than 0(x) • in the fourth case f(x) has an infinity of order or is an infinitesimal of lower order than ¢ (x). For analytic functions the notion of order is well established, for other functions not.
Discontinuous Functions.—A function may fail to be continuous at a point in several ways. In the first case, Lf(x) may exist but be dif ferent from f(a). In this case the discontinuity is called removable (hebbar), because by modi fying the value of f (a) to be the same as Lf(x) a continuous function is obtained. If Lf(x) fails to exist, f(x) may still approach definite limits from above and from below. These limits are usually indicated by f (a + 0) and f(a) may coincide with either of them or differ from both. All these cases are referred to as discontinuities of the first kind.' 0) or f(a— 0) fails to exist, that is, if f (x) approaches more than one value as x approaches a from the right or from the left, f(x) is said to have a discontinuity of the second kind at x a. For this to occur, f(x) must have an infinity of oscillations in every neighborhood of Such a function is sin 1 -- in the neighborhood of x = 0.
It is easily provable that there is a finite or infinite greatest value approached by f(x) on either side of x=a and likewise a finite or in finite least value approached. These four num bers are generally denoted by 7(a + 0), f(a—p), f(a + 0), f(a — 0), The differences among these four numbers and f(a) give a variety of ways of defining a measure of the amount of discontinuity at a point. For example, the lower bound of the oscillation of f(x) on every interval which includes x is the difference between the greatest and the least of these five numbers and is called the oscillation of f(x) at In classifying discontinuous functions ac cording to the distributions of the points of discontinuity over an interval there are two cases of which other cases are compounds. On the one hand, are the totally discontinuous func tions, i. e., functions discontinuous at every point such, for example, as the function which is 1 if x is a rational number and-1 if x is an ir rational number. On the other hand every in terval may contain a point where f(x) is con tinuous; such a function is called point-wise discontinuous, though there is some divergence of usage in this particular.' A point-wise dis continuous function has the property that the points where the oscillation is greater than any fixed positive number, e, constitute a nowhere dense closed set. On considering an infinite e 8 sequence e, 2 4 . . . of such numbers, it is evident that the set of points of discontinuity of a point-wise discontinuous function is an enumerable set of nowhere-dense closed sets.
The theory of discontinuous functions has received perhaps its main impetus from the study of definite integrals. (See CAwcrtus). The integral of f(x) between a and b, written h=n9-1 : f(x)dx, stands for — limb n a — f(xk), where n xk lies between a + — (b—a) and — (b—a).
The function is said to be integrable if this limit exists and is unique. It is easily seen by aid of the principle of uniform continuity that every continuous function is integrable and it is also evident that a totally discontin uous function is not integrable. Riemann proved that a necessary and sufficient condi tion for the integrability of a discontinuous function is that the set of points at which the oscillation of f(x) is greater than any positive number shall be of content zero. Thus an in tegrable function is continuous or so point wise discontinuous that the set of points of dis continuity is an enumerable set of sets of con tent zero.
The content of a point-set is the definite in tegral of a function which is equal to unity for all points of the set and to zero for all other points. Corresponding to the Borel-Lebesgue generalization of content (see Measure and Measurable Assemblages in ASSEMBLAGES, GENERAL THEORY OF) is a generalization of the definite integral, due to Lebesgue,' which ex tends the notion to a very broad class of func tions. This generalization has been guided largely by the aim to state the definitions of differentiation and integration in such a way that they shall be inverse operations. Accord ing to the usual definitions (see CALCULUS) this is true for a limited class of functions, but there exist, on the one hand, functions whose integrals cannot be differentiated and, on the other hand, functions whose derivatives cannot be integrated. The only effective generaliza tion of the derivative that has been suggested in this connection is the substitution for the regular limit of [f (x)—f (xi))/[x — xi], one of the upper and lower bounds of the values ap proached by it as x approaches x,, from above or below.
Attempts to generalize the limiting process in various directions_ are characteristic of the present tendencies in analysis. The principal results so far have been obtained in connec tion with infinite divergent series by Cesar°, Mittag-Leffier, Borel" and others. Another class of problems is the solution of functional equations. For example, it is known that among continuous functions, if f (x + y)= f (x) f(y), f (x) = ax + b. But is this true if the restriction of continuity be removed? Functions of More than One Variable.— The above account has dealt entirely with func tions of one variable. A single-valued function of several variables, f (xs, x,... xn) is a variable y such that to every set of values (xi, x„) there corresponds one, and only one, value of y. The theory of such functions differs from that which we have been considering mainly in that the point-sets involved are no longer linear, but, rather, n dimensional. The approach to a point in a limiting process may now take place not only from two sides but by an infinity of paths. In grasping this situation, the principal idea so far advanced is that of uniform convergence (cf. SERIES). But on this subject the reader is referred to the .treatises cited below.