Volume 1 / Part 5 — *234. CONTINUITY OF FUNCTIONS.
Reading edition
*234. CONTINUITY OF FUNCTIONS.
Summary of *234.
In the present number we are concerned with the definition and analysis of the continuity of functions. The following definition of continuity is given by Dini[17]:
"We call it [the function] continuous for , or in the point , in which it has the value , if, for every positive number , different from 0 but as small as we please, there exists a positive number , different from 0, such that, for all values of which are numerically less than , the difference is numerically less than . In other words, is continuous in the point where it has the value , if the limit of its values to the right and left of is the same and equal to ...."
By the second form of the above definition, the function of previous numbers is to be called continuous at the point if The first form of the definition can also be so stated as to be free from any reference to number, and derivable from the ideas dealt with in the previous numbers of the present section. For this purpose, instead of "a positive number " we take an interval in which is contained, say . Similarly the "values of which are numerically less than " are replaced by arguments in a certain interval containing .
By *233·423, if the limits of the function as the argument approaches are to be all equal, must not be the maximum or minimum of . We therefore take the interval containing to be an interval in which the end-points are not included, say . Thus our definition becomes
We require further, what is tacitly assumed in Dini's definition, that is a member of which has no immediate predecessor or successor, i.e.
In order to deal more easily with the above definition, we analyse it into the product of four factors, which concern respectively and , and , and , and . In the first place, it is obvious that (A) is the product of and a factor obtained by substituting for in (). If , and , () is the product of and a factor obtained by writing for and for in (); and in virtue of , () becomes i.e. if is transitive,
Hence the function is continuous for the argument a if a satisfies () and the three other hypotheses resulting from replacing by , or by , or and by and . If we substitute for , and for , () becomes
Hence continuity can be studied by studying the hypothesis (), and replacing by and by .
The hypothesis () is an interesting one on its own account. We put Thus "" means that is a member of the value-series such that, if is any later member, the function ultimately becomes less than . If we put further then, if is a member of , the function ultimately becomes less than any later member of , and greater than any earlier member. Hence is the limit of the function as the argument increases indefinitely. Hence, if we substitute for , and if , is the limit of the function as the argument approaches a from below, i.e. (This is proved in *234·462.) Hence, putting in place of , the function is continuous from below at the point if and is continuous from above if These results, and various others connected with them, are proved below. The equivalence of Dini's two definitions is proved in *234·63. It will be observed that practically nothing in the theory of continuous functions requires the use of numbers.
We use the symbol "" for the class of arguments for which the limit of the function for approaches to a from below is . Thus, in virtue of what was said above, we may put Then a function is continuous at the point if a belongs to the two classes and . Hence we put The function is continuous with respect to and if it is continuous for all arguments in . Thus we put
Our propositions in this number begin with the properties of and . We have
*234·103.
Thus the hypothesis enables us to use propositions of previous numbers having the hypothesis .
The identification of our definitions with the usual definitions of continuity of functions proceeds by means of the proposition
*234·12.
We have a collection of propositions dealing with the relations of to and . is an upper section of (*234·131); is the complement of , i.e. of without its maximum (if any). This is expressed in the following proposition:
*234·174.
We thus arrive at
*234·182.
Thus is contained in (*234·201), and therefore has not more than two terms (*234·202). If has one term, this is the only member of (*234·203). If has two terms, they have the relation (*234·242); hence if is a compact series, and is not null, its only member is both and (*234·25), while conversely, if and are equal, each is the only member of (*234·251).
We now apply the above results to the limits of a function as its argument approaches the limit of a class . This is done, as before, by substituting for . We arrive at the proposition (*234·33) that if has Dedekindian continuity, and is not null, its only member is both and , i.e. is the limit of the function as the argument increases in .
We then take for the particular value , so that we become concerned with what happens when the argument approaches a from below. For the comparison of our definition of continuity with such definitions as the one quoted from Dini above, we have
*234·41.
I.e. if is neither the first nor the last member of the -series, belongs to when, and only when, given any interval , however small, in which is contained, there is an argument earlier than , such that the value of the function for all arguments earlier than a but not earlier than lies in the interval .
We deduce from previous propositions that, with the usual hypothesis as to , if is a Dedekindian series, and if is a series and is a unit class, its only member is both and , i.e. is the limit of the function for approaches to from below (*234·43). The following proposition sums up our results:
*234·45.
Thus is, in a compact series, the necessary and sufficient condition for the existence of a definite limit of the function as the argument approaches from below.
Without assuming , if is a member of , and if has no immediate predecessor or successor, so that in the neighbourhood of the series is compact, we still have (*234·462).
We next consider . By the definition we have
*234·5.
Thus is an argument for which the function has a single value which has no immediate predecessor or successor in , and which, in virtue of *234·462, is the limit of the function as the argument approaches from below (*234·52). The cases when or require special attention; excluding these cases, we arrive at
*234·51.
This proposition is analogous to *234·41.
We prove (*234·562) that if , are series, and is any class of arguments for which all the values belong to , and if has a limit at which the function is continuous from below, then the limit of the function, as the argument increases in , is the value of the function at the limit of .
We next consider , which is defined as . We show that if is a series whose field contains and is transitive, and is connected, and is neither nor , then if belongs to the class , is the limit of the function for the argument for approaches either from below or from above (*234·62). If is compact, the converse also holds (*234·63). Our definition of a point of continuity is thus identified with the second form of Dini's definition quoted above. It is identified with the first form by the following proposition: In the circumstances of *234·62, if , we have (*234·64) i.e. is a point of continuity when, and only when, the value for the argument is a member of the -series having no immediate predecessor or successor, and if is contained in the interval , then, however small this interval may be, two arguments , can be found such that a lies between them, and the values for all arguments from to (both included) lie in the interval .
We end with a few propositions on continuous functions. The last of these (*234·73) states that, if is a compact series and is transitive and connected, then is continuous with respect to and when, and only when, it has arguments in , and for all such arguments we have i.e. the value for every argument is the limit for that argument for approaches either from above or from below.
*234·01.
*234·02.
*234·03.
*234·04.
*234·05.
*234·1.
*234·101.
Dem.
*234·102.
Dem.
*234·103.
Dem.
*234·104.
Dem.
*234·105.
Dem.
When , the above proposition is not necessarily true: it may fail if .
It is to be observed that and are functions of , so that they are unchanged when is substituted for . Hence the hypothesis is as effective, with regard to them, as the hypothesis . This is stated in the following proposition.
*234·106.
*234·107.
Dem.
*234·11.
*234·111.
Dem.
*234·12.
Dem.
*234·121.
*234·122.
*234·13.
Dem.
*234·131.
Dem.
*234·14.
Dem.
*234·141.
*234·142.
Dem.
*234·15.
Dem.
*234·16.
*234·161.
Dem.
*234·162.
Dem.
*234·17.
Dem.
*234·171.
Dem.
*234·172.
Dem.
*234·173.
*234·174.
Dem.
*234·175.
*234·18.
Dem.
In virtue of this proposition, and are complementary sections of , i.e. they constitute a Dedekind cut in .
*234·181.
Dem.
*234·182.
Dem.
*234·183.
*234·2.
Dem.
*234·201.
*234·202.
*234·203.
*234·204.
*234·21.
Dem.
*234·23.
Dem.
*234·24.
Dem.
*234·241.
Dem.
*234·242.
Dem.
*234·243.
Dem.
*234·244.
Dem.
*234·25.
*234·251.
Dem.
*234·26.
*234·27.
Dem.
*234·271.
*234·272.
The remaining propositions of the present number are for the most part immediate consequences of those already proved. In order to obtain, from propositions already proved, propositions concerning the limit of a function as the argument approaches the limit of some class of arguments , we only have to substitute for . In order to obtain the limit of a function as the argument approaches a given term , we take in place of .
*234·3.
*234·301.
*234·31.
*234·311.
*234·312.
*234·32.
*234·321.
*234·322.
*234·329.
*234·33.
*234·331.
*234·34.
*234·35.
*234·351.
*234·352.
*234·4.
*234·41.
*234·42.
*234·421.
*234·422.
*234·43.
*234·439.
*234·44.
*234·441.
*234·45.
*234·46.
*234·461.
*234·462.
*234·5.
*234·51.
Dem.
*234·52.
*234·521.
*234·522.
Dem.
*234·53.
Dem.
*234·54.
Dem.
*234·55.
*234·56.
Dem.
*234·561.
*234·562.
That is, if is any class of arguments having a limit at which the function is continuous, then the limit of the function, as the argument approaches the limit of the set of arguments, is the value of the function for that limit.
*234·6.
*234·61.
Dem.
*234·62.
*234·63.
*234·64.
*234·7.
*234·71.
Dem.
*234·72.
*234·73.
Dem.
FOOTNOTES:
CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS