Functions of Real Variables I

FUNCTIONS OF REAL VARIABLES I. The Development of the Theory of Functions goes back to Descartes' publication of his work on analytic geometry (1637). The abscissas x and ordinates y of a plane curve are variables; the curve pictures the dependence of the one on the other. The word function seems first to have been used in such a sense by Leibniz (1694), who thus denoted certain lengths such as abscissas, ordinates, tangents, normals, radii of curvature, associated with the position of a point on a curve. J. Bernoulli (1698) applied the term to mathematical expressions involving variables and constants. The notation f(x) was used by L. Euler in 1734. His Introductio in Analysin In finitorunr (1748) may be regarded as the first treatise on the theory of functions, although its point of view would now be described as elementary and formal. As in the definition of Bernoulli, a function is identified with its analytic expression and according to the form of this expression it is classi fied as explicit or implicit, algebraic or transcendental.

The eighteenth century was a constructive period in mathe matics, an epoch of discovery, when scientific accuracy played a subordinate role. The first part of the nineteenth century ushered in the present era of criticism. The Theorie des fonctions analy tiques of J. L. Lagrange (17 97) was a precursor of this change. Better foundations were here sought by basing the processes of the calculus on the properties of series of powers of the independ ent variable. Analytic expressions were still of central interest, but soon a new point of view was forced upon mathematicians by the discovery of J. B. J. Fourier (1807) that a single analytic ex pression, a trigonometric series, may represent, in different do mains, what had been regarded as different functions. It was be lieved at first that an arbitrary function could be represented by a series of Fourier. When it appeared that there were necessary restrictions, the way was open to a treatment of functions based on their intrinsic properties. In connection with his proof of the convergence of Fourier series (1829) P. G. Lejeune-Dirichlet gave essentially the definition of a function that appears at the head of this article (Werke, vol. 1, p. Among the founders of the modern theory of functions, both real and complex, three are pre-eminent, A. L. Cauchy (1789 1857), G. F. B. Riemann (1826-66), and K. Weierstrass (1815 97). To the latter is especially due the arithmetization of the sub ject, whereby all definitions are based on equalities and inequalities concerning numbers, and geometrical notions are avoided. The theory of aggregates (q.v.), which furnishes a necessary founda tion for ideas regarding domains, and which arose from the in vestigations of G. Cantor (1845-1918), has become one of the most important adjuncts of the theory of functions. Its develop ment has been connected with recent generalizations of the notion of integration by H. Lebesgue (1902) and others.

2.

The Real Variable and its Domain.—The domain of a real variable is a set of real numbers, which may be positive or negative integers, rational numbers (fractions) or irrationals. In accordance with a fundamental postulate, such sets may be put into one-to-one correspondence with points on a line-segment, hence the point thus associated with a number is often substituted for it in the terminology of the subject. Domains are then re ferred to as point sets (q.v.), and the domain of a single variable is said to be a linear point set or aggregate. Since in most cases the domain of a variable is an infinite set of numbers, a general theory of such aggregates and of their properties is fundamental. We shall not here attempt to describe that theory, for an account of which the reader is referred to the article on NUMBER.

The most important domains for a real variable x are intervals consisting of all real numbers between given numbers a and b, or, in geometric language, of all points on an x axis between the point x = a and the point x = b. Such an interval is denoted by the symbol [a, b] . It is closed if it includes the values x = a and x= b, and is open if it does not include them. A neighbour hood of a point x' is an interval which includes that point. A do main has an upper bound if its numbers are all less than a fixed number N; it has a lower bound if its numbers are all greater than a fixed number M; when one of these bounds does not exist, the domain is said to be unbounded or unlimited in that sense. For functions of more than one variable (see § 8 below), the notion of region corresponds to that of interval for a single variable.

A point x' is a limit point of a domain if in every neighbour hood of that point there is a point of the domain other than x'; the point x' may or may not itself belong to the domain. Every bounded set of points whose number is infinite has at least one limit point, but this is not true, for example, of the set of all positive integers, which has no upper bound. This set is, however, said to have the limit + oo (positive infinity) and similarly a set without a lower bound has the limit — oo.

A function is defined when, in addition to specifying the domain of the independent variable x, we give a rule which will serve to compute the corresponding values of the dependent variable y. Such a rule may be, and commonly is, given by an analytic ex pression in x whose value for each x of the given domain is to be taken as the value of y ; or different expressions may serve for dif ferent parts of the domain of x. Such expressions represent the function for the corresponding values of x.

We shall now consider some of the basic notions of the theory of functions of a real variable. To supplement this necessarily brief introduction to the subject, the reader should consult the authorities cited at the end.

3. Upper and Lower Bounds: Limits.—For a given function y= f (x) defined on a given domain of values of x, the values of y also form a domain. If these values have an upper or a lower bound the same is said to be true of the function. In particular, if a function defined for a given domain of x has an upper bound. it has a least upper bound which is either, (I) a value of y not exceeded by any other value of y, but not a limit point of the y domain, or (2) a limit point for the values of y which is not ex ceeded by any of those values, but which may or may not itself be a value of y. In case (2) the least upper bound is the upper limit of the function, i.e., the least upper bound of the limit points of y. In these definitions, whose extension to lower bounds is obvious, we may restrict the domain of x to a part of the total domain of definition of the function. As an illustration of these terms it is to be noted that the function represented by the ex pression has for its least upper bound, which is also the upper limit, on the open interval [o, 1 ] of the x axis the value y =I, but this value is not attained by the function on that interval, since the end-point x = I is not included. The difference between the least upper bound and the greatest lower bound of a function in a domain is called the oscillation or fluctuation of the function for that domain.

In order to describe the variation of a function on a given domain, we style f(x) monotonic increasing if for every sequence of increasing x values the corresponding values of f(x) always increase; similarly a function is monotonic decreasing if its values decrease as x increases. The least upper and greatest lower bounds of such functions correspond to end values or limits of the x domain. Oscillating functions are non-monotonic ; an im portant class, those of limited variation, is composed of functions representable as differences of monotonic functions.

The properties specified so far are related to the behaviour of a function throughout a domain. The most important notion connected with the neighbourhood of a point, x= X, is that of the limit of f(x) as x approaches X. We suppose f(x) a single valued function on a domain for which X is a limit point, but of which X is not necessarily a member. By a 3-neighbourhood of X we designate all x's of the domain, except X itself, that lie between X-6 and X+6. Let e be the fluctuation of f(x) on a 3-neighbourhood of X. If e can be made arbitrarily small by making b a sufficiently small positive number, f(x) is said to approach a limit as x approaches X. It can be proved that there then exists a number L such that no matter what positive number E is preassigned the numerical value of f(x) —L is less than E for all values of x in a 6-neighbourhood of X. This number L is the limit (x) as x approaches X, a relation whose symbolic expression is lim f (x) = L.

The definition of a limit is extended to unbounded domains, with the notation f (x) = L, if for every positive number E a number N exists such that the numerical value of f(x) — L is less than E for all values in the domain of x that are greater than N. There are similar definitions for a limit as x becomes nega tively infinite, and for a function's becoming positively or nega tively infinite. The sum of an infinite series furnishes an impor tant application of these definitions. Consider, for example, the series 2m4m . .. . . . , of which the sum of n terms is s„ =2-1--4+ . . . = I —2n a function of the real vari able n on the domain of all positive integers, n= i, 2, 3 . . . . The numerical difference of s,, and i is eft, and this can be made less than any preassigned positive number e for all values of n greater than any positive number N for which 2 N is less than E. Thus I, and this limit is, by definition, the sum or value of the series.

Various extensions of the notion of limit play a considerable role in the modern theory of functions. For example, although f(x) may not have a limit as x approaches X when all values of the domain are considered, there may be partial domains for which a limit exists. In fact it can be proved that if X is a limit point of a domain, then there is always at least one set of values belonging to this domain on which f (x) either approaches a limit or becomes infinite as x approaches X. Restricted domains of especial interest are those whose values are all greater than X, or all less than X. For a domain of the former sort, for example, we write f (x) =L, or f (X +o) =L. If f(x) is bounded and mono tonic on such a domain, f(X+o) always exists.

4. Continuity.

It is to be noted that in the definition of a limit we have not assumed that L is the value of f(x) for x= X, or even that f(X) exists. A function defined on an interval [a, b] including the point X is continuous at X if it has the limit f(X) as x approaches X on the interval. An extension of this definition styles f(x) continuous on the right at X if f(X+o) = f (X) ; we similarly describe continuity on the left. A function is continuous throughout an interval [a, b] if it is continuous at every point of the interval. If [a, b] is closed, it is here implied that f(a) = f(a+o), and f (b) = f (b—o).

Sums, differences, products and quotients of functions con tinuous on an interval are also continuous there, with the excep tion that a quotient may not be continuous for values which make its denominator equal to zero. More generally, it may be stated that every function of a continuous function is continuous in the sense that if f(x) is continuous at X, and if F(y) is defined throughout a neighbourhood of y = f (X) and is continuous at f(X), then F [f(x)] is continuous at X.

Since a continuous function [for example, f(x) =xsini/x] may oscillate infinitely often in the neighbourhood of a point, it is obvious that a complete graphic representation of all such functions is not possible. On the other hand all functions are continuous whose graphs as drawn in a system of ordinary rec tangular coordinates are unbroken curves of one piece over an x interval. If this interval [a, b] is closed, a function thus graphic ally defined is evidently bounded on [a, b] and can easily be shown to have maximum and minimum values there. Further, every length between those of f(a) and f(b) will fit as an ordi nate of the curve y = f (x) somewhere between x = a and x= b. The corresponding properties are shared by all functions that are continuous on a closed interval. Another property of all functions continuous on a closed interval [a, b] is that of uniform continuity, according to which it is always possible to subdivide [a, b] into a finite number of parts such that in each the fluctua tion of the function is less than a preassigned positive number. On this property rests the proof that every function continuous on a closed interval [a, b] has a definite integral from a to b (see 6 below).

A function continuous on [a, b] is completely determined by its values at all rational points of [a, b], or by its values at any other set of points everywhere dense on [a, b], i.e., a set whose points are to be found in every subinterval of [a, b]. Since the rational points of an interval are enumerable, that is, can be put into one to-one correspondence with the set of all positive integers, it fol lows that a continuous function can be completely defined by an enumerable number of conditions.

5. Discontinuous Functions.

A function defined on a neigh bourhood of X, but not necessarily at X itself, is discontinuous at X if any part of the definition of continuity is not satisfied. This will be the case, in particular, if the function is not defined at X. An infinite discontinuity occurs if the function becomes infinite as x approaches X, either on the whole interval, or on some set of points of which X is a limit point. If the function is bounded on a neighbourhood of X but is not continuous at X, it has a finite discontinuity at that point. One class, that of removable discon tinuities, consists of those presented by functions which have a limit L as x approaches X on a neighbourhood of X, but either f(x) is not defined at X or f (X) is not equal to L. When f(X) is suitably defined the function becomes continuous at X. Another variety of finite discontinuities is presented when f(X+o) and f(X—o) each exist, but these two limits are not equal. In other cases the function may oscillate so that its fluctuation in every neighbourhood of X is greater than a positive constant [this is, for example, true of f (x) = sin i/x at the origin] . Here the greatest and least limits approached by f(x) on all sets of points to the right of X and for which X is a limiting point, together with similar limits on the left [designated f (X ±o), f(X+o), f (X—o), f(X—o), respectively], furnish a measure of the oscillation of f(x) at X. Examples can be given of functions that are discontinuous at every point of an interval, and of others that are continuous at every irrational point and discontinuous at every rational point.

6. Derivatives and Integrals.

The differential calculus is based on the idea of the derivative of a function f (x) at a point X, (see CALCULUS). The derivative is, by definition, When X is considered as a variable, f'(X) is said to be the derived function of f(X), but the term derivative is also used with this meaning. It is the main business of the differential calculus to investigate formulas for the derivatives of various analytic expressions and to make application of these formulas. The theory of functions concerns itself with conditions for the existence of the derivative and with generalizations of its defini tion. A function continuous at a point does not necessarily have a derivative there ; Weierstrass has given an example of a func tion continuous at every point of an interval but without a deriv ative at any point whatever. If, however, a function defined and single-valued throughout a neighbourhood of X has a deriva tive at X, then it is continuous at that point.

Various generalizations of the notion of a derivative have been found useful, particularly those in which the limit entering into the definition is taken over restricted domains for x. Thus the limit F(X+o) is designated the progressive derivative of f(x) at X, and F(X—o) the regressive derivative; F(X -}-o), F(X +o), F(X—o), F(X—o) are upper and lower extreme derivatives on the right and left respectively. The derivative of the derivative is called the second derivative, and the process may be carried on to derivatives of the nth order. From the existence of the derivative of order n—i it does not follow that there is a derivative of nth order, or, if the latter exists, that it is continuous.

The integral calculus had its source in problems such as that of determining the area under a curve, for which a solution is obtained by subdividing the region under consideration into parts that are approximately rectangular and taking the limit of the sum of the partial areas. This led to the definition of the integral of f(x) from a to b by means of the formula into n successive subintervals of lengths ..., Oxn, all less than /ix, and that . . ., x„ are points chosen arbitra rily in the respective subintervals corresponding to their subscripts. The integral thus defined is called the definite integral, to dis tinguish it from an indefinite integral or primitive function of f (x), i.e., a function 4 (x) whose derivative is f (x). The two notions are brought together by the theorem that if f(x) is integrable, that is, has a definite integral from a to b, and if x is between a and b, then an indefinite integral 4 (x) can differ at most by a constant from the definite integral of f(x) from a to x. Every function continuous on a closed interval [a, b] is integrable from a to b.

The integral as thus defined is called the Riemann integral since its precise formulation was given in the famous memoir of Riemann on trigonometric series (1854). A necessary and suffi cient condition that a function bounded on a closed interval [a, b] be integrable from a to b can be given in terms of the notion of content. The content of a set of points in an interval [a, b] may be defined as the integral from a to b of a function equal to 1 at each point of the set, and to o at all other points. The condition for integrability is that the points of discontinuity of f(x) on [a, b] form a set whose content is zero.

More general definitions of integration, especially that due to Lebesgue, have had a very considerable influence on recent pro gress in the theory of functions. These generalizations are largely based on extensions of the notion of content; one of their aims has been to frame definitions so that integrals shall be differen tiable and derivatives integrable for broad classes of functions. Another line of investigation has concerned itself with improper integrals, those in which either the function becomes infinite be tween the limits of integration, or the latter are not both finite.

7. Types of Analytic Representation.

In elementary an alysis it is customary to identify functions with analytic expres sions. According to a familiar classification the elementary func tions are polynomials, of form . +a,,; rational functions, quotients of polynomials; power functions, of form x"; algebraic functions, given by equations f(x, y) = o where f(x, y) is a polynomial in both x and y; transcendental functions, such as the exponential function a-r, the logarithmic function log x, trigo nometric and inverse trigonometric functions such as sin x, sin and functions defined by a finite number of functional operations within this class.

Limits of sets of elementary functions furnish a more general class of analytical representations. These may take the form of infinite series or products, or definite integrals. Representations by infinite series are of especial importance. An infinite series of functions of x defined on [a, b], ... ... , is said to converge for x= X, if the sum of n terms has, for x= X, a limit when n becomes infinite. This limit is the value of the series for x= X. A function is represented by a convergent series of functions on [a, b] if the value of the function at each point of (a, b) is the value of the series at that point. Among the series most studied have been the power-series, of type . . . , and trigonometric series, whose nth terms are of type a„cos nx+ b„sin nx. Infinite series that are not convergent may still, in a sense, represent a function. Hence such divergent series have also been the objects of much research.

8. Functions of More Than One Variable.

Many of the properties of functions of one variable can be readily generalized for functions of two or more variables. As the geometry of several dimensions is, however, far more complicated than that of one dimension, it is clear that new notions must be introduced. Modern investigations have not stopped with functions of finitely many variables, but have gone on into domains of infinitely many dimensions.