Volume 1 / Part 5 — *233. ON THE LIMITS OF FUNCTIONS.
Reading edition
*233. ON THE LIMITS OF FUNCTIONS.
Summary of *233.
There are four limits of a function as the argument approaches some term in the argument-series, namely the upper and lower limits of the ultimate oscillation for approaches from below and above respectively. If the ultimate oscillation for approaches to from below reduces to a single term, i.e. if , that one term is the limit of the function for approaches to from below. If this one term is also the ultimate oscillation for approaches from above, we may call it simply the limit of the function for the argument . This may or may not (when it exists) be equal to the value for the argument . It is characteristic of continuous functions that the limit exists for every argument, and is always equal to the value for that argument. Continuous functions will be considered in *234.
The upper limit or maximum of the ultimate oscillation as the argument approaches is the upper limit or maximum of the ultimate section. Hence if we put the four limits of the function as the argument approaches will be It will be seen that is a function of . It may happen that, if we put in place of , the function will have a definite limit as the argument increases in , although has no limit or maximum. Thus if, for example, consists of the series of rationals, and of the series of real numbers, if is a class of rationals not having a rational limit, we may regard the limit of the function (if it exists), as the argument increases in , as the value of the function for the irrational limit of . In this way we can extend the domain of definition of a function. In order to be able to deal with the cases in which has no limit, we put If is a Dedekindian series, always exists. If we take to be any segment of , we thus get a new function, derived from , but having segments of instead of members of as its arguments. Thus if had rationale for its arguments, this new function will have real numbers for its arguments. (Real numbers may be regarded as segments of the series of rationals.)
The function is a particular case of the above; thus we take as our definition or, what comes to the same thing,
The following propositions of this number are important:
*233·15.
*233·16.
*233·2—·25 are applications of the more important of the propositions *232·34—·39, showing circumstances under which the limit of the function for the class is the same as for the class .
*233·4 and following propositions apply the earlier propositions of *233 to the case where is replaced by , and therefore is replaced by . We have
*233·43.
*233·433.
*233·45.
I.e. in a series having Dedekindian continuity, the necessary and sufficient condition that the two limits of the function as the argument approaches from below should be equal is that the ultimate oscillation should not have more than one term.
We have next a set of propositions (*233·5—·53) on the possibility of replacing by a class having for its limit, without altering the limits of the function. We have to begin with
*233·5.
in virtue of *207·291. Thence by earlier propositions of this number,
*233·512.
whence we obtain
*233·514.
Thus if , are series, and is the limit of the function for the argument ( being a term which has no immediate successor or predecessor), is the limit of the function for any class of arguments whose limit is . Hence we arrive at the proposition
*233·53.
Thus if has Dedekindian continuity, and is a class of arguments having a limit, and if the ultimate oscillation as the argument approaches this limit has not more than one term, the limit of the function for the class exists, and is equal to the limit of the function for the argument .
*233·01.
*233·02.
*233·1.
*233·101.
*233·102.
*233·103.
*233·11.
*233·111.
*233·12.
Dem.
*233·13.
*233·14.
*233·141.
*233·142.
Dem.
*233·15.
*233·16.
Dem.
*233·17.
Dem.
*233·171.
Dem.
*233·172.
Dem.
*233·173.
*233·174.
Dem.
*233·2.
*233·21.
*233·22.
Dem.
*233·23.
*233·24.
*233·241.
*233·25.
*233·4.
*233·401.
*233·402.
*233·41.
Dem.
*233·42.
*233·421.
Dem.
*233·422.
*233·423.
*233·424.
*233·425.
*233·426.
*233·43.
*233·431.
*233·432.
*233·433.
*233·434.
*233·435.
*233·44.
*233·45.
*233·5.
*233·501.
Dem.
*233·51.
*233·511.
*233·512.
*233·513.
*233·514.
*233·515.
*233·516.
*233·52.
*233·53.