Volume 1 / Part 5 — SECTION C. ON CONVERGENCE, AND THE LIMITS OF FUNCTIONS.

The purpose of this section is to express in a general form the definitions of convergence, the limits of functions, the continuity of functions, and kindred notions, and to give such elementary consequences of these definitions as may seem illustrative.

In the definitions usually given in treatises on analysis, it is assumed that both the arguments and the values of the function are numbers of some kind, generally real numbers, and limits are taken with respect to the order of magnitude. There is, however, nothing essential in the definitions to demand so narrow a hypothesis. What is essential is that the arguments should be given as belonging to a series, and that the values should also be given as belonging to a series, which need not be the same series as that to which the arguments belong. In what follows, therefore, we assume that all the possible arguments to our function, or at any rate all the arguments which we consider, belong to the field of a certain relation , which, in cases where our definitions are useful, will be a serial relation; we assume similarly that the values of our function, at least for arguments belonging to , belong to the field of a relation , which, in all important cases, will be a serial relation. The function itself we represent by the relation of the value to the argument; that is, the relation of to is to be , so that, if the function is one-valued, . (If the function is not one-valued, is any member of .) Thus we may speak of as the function, as the argument-series, and as the value-series.

To take an illustration: Suppose we are given a set of real numbers , , ... , ..., where may be any finite integer. Here is a function of ; the argument-series is that of the finite integers in order of magnitude, the value-series is that of the real numbers (or any part of this series which contains all the values , , ... , ...). The function is the relation of to , so that . In this case, calling the argument-series and the value-series (as will be done throughout this section), we have , , , ... , ..., and = the series , , ... , .... The series which arranges , , ... , ... in the order of their own magnitudes, instead of the order of magnitude of their suffixes, is or . This will not be equal to unless the function is one which continually increases, i.e. one for which .

In general, the propositions of the present section are only important when and are series. If our assertions are not to be trivial, we must have and , i.e. there must be arguments in which lead to values in . It will also generally happen that the function is one-valued, i.e. that . But the above conditions, though necessary to the importance of our propositions, are in general much narrower than the hypotheses that are necessary for the truth of our propositions.

The present section is wholly self-contained, that is to say, its propositions are not referred to in the sequel. We have, in this section, carried the subject as far as seemed suitable for the present work; its further development belongs to treatises on analysis.

We begin (*230) with a general conception which is involved in the notion of convergency. We shall say that the values of a function converge (or, simply, that the function itself converges) into the class , if for late enough arguments the values always belong to the class , i.e. if there is a term such that, if , , or, to avoid assuming that is one-valued, . Thus the values of the function converge into the class if If a term is one such that, from onward, all values belong to , we write (where "" stands for "convergent"), i.e. we put When there is such a , i.e. when the function converges into the class , we write "," i.e. we put "" may be read " is -convergent into ." This means that for arguments sufficiently late in the -series, the value of the function is always a member of . Thus e.g. if , and , , and if , .

We next consider (*231) limiting sections and ultimate oscillations of functions. For this purpose, we proceed as follows. If , then is a section of the -series such that, for sufficiently late arguments, the values of the function must belong to . Hence if we take all possible values of for which , and take the logical product of all the resulting sections , we get a section containing all the "ultimate" values of the function; moreover this is obviously the smallest section which has this property, because, if we take any section which contains all the "ultimate" values, we have , and , and therefore the logical product in question is contained in . The logical product in question is In order to avoid trivial exceptions which arise when , we define the "limiting section" as This "limiting section" we denote by , where the letters "" stand for "section." Thus we put

is the class of those members of the series which are such that, given any argument however late, there are still arguments as late or later for which the value of the function is not less than . In like manner, , which we will call the "limiting upper section," consists of those members of the series which are such that, given any argument however late, there are still arguments as late or later for which the value of the function is not greater than . Thus the product of and is the smallest stretch which contains all the "ultimate" values of the function, i.e. it is the stretch consisting of those terms which are such that, however late an argument we take, there are arguments as late or later for which the value of the function is not greater than , and also arguments for which it is not less than . Thus the product of and represents what we may call the "ultimate oscillation" of the function. We shall denote it by , putting We may express in a form not involving , namely (*231·12)

This formula for may be elucidated by the following considerations. If is any member of , then consists of all arguments from onwards. Hence . i.e. , consists of all values of the function for arguments from onwards. Hence consists of all members of the -series which are equalled or surpassed by values of the function for arguments equal to or later than . Now if a term belongs to the class for every argument , it is a term such that, however far up the argument-series we go, we shall still find values as great as or greater than . When this is the case, we may say that is -persistent. In this case, may be regarded as not greater than the "ultimate" values of the function. Now the class of arguments concerned is . Hence the class of -persistent terms is where the factor may be added in order to accommodate the formula to the trivial case where (the only case in which the factor makes any difference). Thus the class of -persistent terms is the limiting section. Similarly the -persistent terms are the limiting upper section. These are the terms which are not less than the "ultimate" values of the function. Thus the product is the terms which are neither greater than all ultimate values, nor less; hence it is the class of ultimate values, which may be appropriately called the "ultimate oscillation."

It will be seen that , being the product of an upper and lower section, is itself a stretch: we may call it (alternatively) the "limiting stretch." It consists of all members of the -series such that the function does not, however great we make the argument, become and remain less than , nor yet become and remain greater than . If consists of a single term, that term is the limit of the function as the argument travels up the series . (This is, of course, in general different from the limit of the values of the function considered simply as a class of members of , i.e. it is different from .) If does not consist of a single term or none, we shall have two limits to consider, namely and , which give the two boundaries of the ultimate values of the function. When the class is null, the function may be regarded as having a definite limit: in this case, and are the two parts of an "irrational" Dedekind cut, i.e. a cut in which the first portion has no maximum and the second no minimum. Thus is the condition for a definite limit of the function as the argument grows indefinitely.

The above gives the generalization of the limit of a function when the argument may be any member of . In order to obtain limits for other classes of arguments, it is only necessary, as a rule, to limit the field of to the class of arguments in question, i.e. to replace by (cf.*232). In order, however, to avoid vexatious and trivial exceptions arising when , it is more convenient to replace by . Thus the section of defined by the class of arguments is ). We put This definition is useful because we very often wish to be able to exhibit the limiting section defined by as a function of . The section is such that, if is any member of it, and is any argument belonging to , there is in an argument equal to or later than , for which the function has a value equal to or later than . Thus is such that the function does not ultimately become less than as the argument increases in the class . The limit or maximum of such terms as is the limit or maximum of the ultimate values of the function as the argument approaches the top of . The class of ultimate values is If the function has a definite limit as the argument increases in , the class of ultimate values must not contain more than one term.

Our next number (*233) deals with the limit of a function for a given argument. The limit or maximum of the class of ultimate values is not necessarily the value for the limit of . It will be found, however, that, with a suitable hypothesis, the limiting section depends only upon , and if has no maximum, it depends only upon . Thus if and both have the same limit, they define the same limiting section. Hence if is the limit of , the limiting section of is . The upper limit of this is the upper limit of the ultimate values as the argument approaches from below. We put We have thus four limits of the function as the argument approaches , namely If is a continuous function, these four are all equal to ; but in general they are different from each other and from . The subject of the continuity of functions is dealt with in *234. When , each is the limit of the function for the argument for approaches from below. It should be observed that if is defined for a set of arguments which are dense in , i.e. if , then and are defined for all arguments in .