Volume 1 / Part 5 — *232. ON THE OSCILLATION OF A FUNCTION AS THE ARGUMENT APPROACHES A GIVEN LIMIT.
Reading edition
*232. ON THE OSCILLATION OF A FUNCTION AS THE ARGUMENT APPROACHES A GIVEN LIMIT.
Summary of *232.
In the preceding number, we considered the ultimate oscillation of a function when the argument grows without limit. If, in the propositions of the last number, we confine the field of to , where , the ultimate oscillation becomes the ultimate oscillation as the argument approaches from below. If the ultimate oscillation consists of a single term, this is the limit of the function as the argument approaches from below. If, instead of confining the argument to , we confine it to any other class whose limit is , we shall, under a very usual hypothesis, obtain the same value for the ultimate oscillation as if we confined it to . And more generally, under a similar hypothesis, if and are two classes of arguments which define the same section (i.e. such that ), then, whether or not this section has a limit, the ultimate sections and the ultimate oscillation are the same for as they are for . Hence we are led to consider first the result of confining the field of , not to , but to any class . In order not to have to exclude explicitly the case in which , we deal with , not . Hence we are led to the following definitions:
*232·01.
*232·02.
Most of the propositions of the present number are immediate consequences of corresponding propositions in *231. The most important application of the propositions of the present number is to the case where is of the form , being a member of . We may, in this case, take in place of any other class of arguments (e.g. a progression of arguments , , ... , ...) having for its limit, without altering the limiting sections or the ultimate oscillation. Hence the limit of the function for a given argument (if it exists) may be determined by choosing any selection of arguments having the given argument as their limit (cf. *233·142, below).
From the definition of we obtain immediately
*232·11.
We prove that (*232·131), and that if , the two limiting sections and the ultimate oscillation are all equal to (*232·15). Also we have
*232·14.
Thus the substitution of for in our definitions has the effect of making them applicable to unit classes, and of enabling us to substitute the hypothesis for . But when is transitive and connected (and therefore when is a series), the substitution of for in the definitions makes no difference unless is a unit class. This case is trivial, since the only interest of our definitions is when a has no maximum in .
From *231·22 we obtain
*232·22.
We have next a set of propositions concerned in discovering circumstances under which two classes and which determine the same section in (and therefore have the same limit, if any) give the same values for the two limiting sections. For this purpose, it is only necessary to discover circumstances under which we may substitute for . When this can be done, the ultimate oscillation of the function as the argument approaches the limit of a can be determined by taking any set of arguments having this limit. We have
*232·301.
*232·32.
Thus if the function has a limit as the argument approaches the limit of , it also has a limit as the argument approaches the limit of .
*232·33.
whence
*232·34.
We have also
*232·341.
Hence we arrive at the conclusion that, if is a series, and is the limit of the function for the class a , if is a member of , it is its maximum (*232·352), while if is not a member of , it is its sequent (*232·356), assuming , which, as we saw (*233·22), is generally the case, and assuming also . On the other hand, if has no maximum, is the minimum of ; and if has a maximum other than , this is (*232·357·358). This latter case is impossible unless has an immediate predecessor. Hence we arrive at the following proposition:
*232·38.
Applying this to a series having Dedekindian continuity, we know that , and that the and always exist. Hence
*232·39.
That is to say, if the value-series has Dedekindian continuity, and contains all values for arguments in , then, provided the function has a definite limit for the class , this is its limit also for the class ; that is to say, any collection of arguments having the same limit or maximum as a given section will give the same limit for the function.
*232·01.
*232·02.
*232·1.
*232·101.
*232·11.
Dem.
*232·12.
*232·121.
Dem.
*232·13.
*232·131.
From the above propositions it follows that the values of , , and depend only upon ; thus if is not contained in , the part not contained in is irrelevant.
*232·14.
*232·15.
*232·151.
*232·2.
Dem.
*232·21.
*232·22.
*232·23.
Dem.
*232·24.
Dem.
*232·3.
Dem.
*232·301.
Dem.
*232·31.
Dem.
*232·32.
*232·33.
Dem.
*232·34.
*232·341.
*232·35.
*232·351.
Dem.
*232·352.
*232·353.
Dem.
*232·354.
*232·355.
Dem.
*232·356.
*232·357.
Dem.
*232·358.
Dem.
*232·36.
*232·361.
Dem.
*232·37.
*232·38.
*232·39.
Dem.
*232·5.
*232·51.
*232·511.
*232·52.
*232·53.
Dem.