IN this chapter we shall be concerned with the definition of the limit of a function (if any) as the argument approaches a given value, and also with the definition of what is meant by a "continuous function." Both of these ideas are somewhat technical, and would hardly demand treatment in a mere introduction to mathematical philosophy but for the fact that, especially through the so-called infinitesimal calculus, wrong views upon our present topics have become so firmly embedded in the minds of professional philosophers that a prolonged and considerable effort is required for their uprooting. It has been thought ever since the time of Leibniz that the differential and integral calculus required infinitesimal quantities. Mathematicians (especially Weierstrass) proved that this is an error; but errors incorporated, e.g. in what Hegel has to say about mathematics, die hard, and philosophers have tended to ignore the work of such men as Weierstrass.
Limits and continuity of functions, in works on ordinary mathematics, are defined in terms involving number. This is not essential, as Dr Whitehead has shown.[22] We will, however, begin with the definitions in the text-books, and proceed afterwards to show how these definitions can be generalised so as to apply to series in general, and not only to such as are numerical or numerically measurable.
[22]See Principia Mathematica, vol. II. * 230-234.
Let us consider any ordinary mathematical function , where and are both real numbers, and is one-valued—i.e. when is given, there is only one value that can have. We call the "argument," and the "value for the argument ." When a function is what we call "continuous," the rough idea for which we are seeking a precise definition is that small differences in shall correspond to small differences in , and if we make the differences in small enough, we can make the differences in fall below any assigned amount. We do not want, if a function is to be continuous, that there shall be sudden jumps, so that, for some value of , any change, however small, will make a change in which exceeds some assigned finite amount. The ordinary simple functions of mathematics have this property: it belongs, for example, to , , ... , , and so on. But it is not at all difficult to define discontinuous functions. Take, as a non-mathematical example, "the place of birth of the youngest person living at time ." This is a function of ; its value is constant from the time of one person's birth to the time of the next birth, and then the value changes suddenly from one birthplace to the other. An analogous mathematical example would be "the integer next below ," where is a real number. This function remains constant from one integer to the next, and then gives a sudden jump. The actual fact is that, though continuous functions are more familiar, they are the exceptions: there are infinitely more discontinuous functions than continuous ones.
Many functions are discontinuous for one or several values of the variable, but continuous for all other values. Take as an example . The function passes through all values from -1 to 1 every time that passes from to , or from to , or generally from to , where is any integer. Now if we consider when is very small, we see that as diminishes grows faster and faster, so that it passes more and more quickly through the cycle of values from one multiple of to another as becomes smaller and smaller. Consequently passes more and more quickly from -1 to 1 and back again, as grows smaller. In fact, if we take any interval containing 0, say the interval from to where is some very small number, will go through an infinite number of oscillations in this interval, and we cannot diminish the oscillations by making the interval smaller. Thus round about the argument 0 the function is discontinuous. It is easy to manufacture functions which are discontinuous in several places, or in places, or everywhere. Examples will be found in any book on the theory of functions of a real variable.
Proceeding now to seek a precise definition of what is meant by saying that a function is continuous for a given argument, when argument and value are both real numbers, let us first define a "neighbourhood" of a number as all the numbers from to , where is some number which, in important cases, will be very small. It is clear that continuity at a given point has to do with what happens in any neighbourhood of that point, however small.
What we desire is this: If is the argument for which we wish our function to be continuous, let us first define a neighbourhood ( say) containing the value which the function has for the argument ; we desire that, if we take a sufficiently small neighbourhood containing , all values for arguments throughout this neighbourhood shall be contained in the neighbourhood , no matter how small we may have made . That is to say, if we decree that our function is not to differ from by more than some very tiny amount, we can always find a stretch of real numbers, having in the middle of it, such that throughout this stretch will not differ from by more than the prescribed tiny amount. And this is to remain true whatever tiny amount we may select. Hence we are led to the following definition:—
The function is said to be "continuous" for the argument 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[23] than , the difference is numerically less than .
[23]A number is said to be "numerically less" than when it lies between and .
In this definition, first defines a neighbourhood of , namely, the neighbourhood from to . The definition then proceeds to say that we can (by means of define a neighbourhood, namely, that from to , such that, for all arguments within this neighbourhood, the value of the function lies within the neighbourhood horn to . If this can be done, however may be chosen, the function is "continuous" for the argument .
So far we have not defined the "limit" of a function for a given argument. If we had done so, we could have defined the continuity of a function differently: a function is continuous at a point where its value is the same as the limit of its value for approaches either from above or from below. But it is only the exceptionally "tame" function that has a definite limit as the argument approaches a given point. The general rule is that a function oscillates, and that, given any neighbourhood of a given argument, however small, a whole stretch of values will occur for arguments within this neighbourhood. As this is the general rule, let us consider it first.
Let us consider what may happen as the argument approaches some value from below. That is to say, we wish to consider what happens for arguments contained in the interval from to , where is some number which, in important cases, will be very small.
The values of the function for arguments from to ( excluded) will be a set of real numbers which will define a certain section of the set of real numbers, namely, the section consisting of those numbers that are not greater than all the values for arguments from to . Given any number in this section, there are values at least as great as this number for arguments between and , i.e. for arguments that fall very little short of (if is very small). Let us take all possible 's and all possible corresponding sections. The common part of all these sections we will call the "ultimate section" as the argument approaches . To say that a number belongs to the ultimate section is to say that, however small we may make , there are arguments between and for which the value of the function is not less than .
We may apply exactly the same process to upper sections, i.e. to sections that go from some point up to the top, instead of from the bottom up to some point. Here we take those numbers that are not less than all the values for arguments from to ; this defines an upper section which will vary as varies. Taking the common part of all such sections for all possible 's, we obtain the "ultimate upper section." To say that a number belongs to the ultimate upper section is to say that, however small we make , there are arguments between and for which the value of the function is not greater than .
If a term belongs both to the ultimate section and to the ultimate upper section, we shall say that it belongs to the "ultimate oscillation." We may illustrate the matter by considering once more the function as approaches the value 0. We shall assume, in order to fit in with the above definitions, that this value is approached from below.
Let us begin with the "ultimate section." Between and 0, whatever may be, the function will assume the value 1 for certain arguments, but will never assume any greater value. Hence the ultimate section consists of all real numbers, positive and negative, up to and including 1; i.e. it consists of all negative numbers together with 0, together with the positive numbers up to and including 1.
Similarly the "ultimate upper section" consists of all positive numbers together with 0, together with the negative numbers down to and including -1.
Thus the "ultimate oscillation" consists of all real numbers from -1 to 1, both included.
We may say generally that the "ultimate oscillation" of a function as the argument approaches from below consists of all those numbers which are such that, however near we come to , we shall still find values as great as and values as small as .
The ultimate oscillation may contain no terms, or one term, or many terms. In the first two cases the function has a definite limit for approaches from below. If the ultimate oscillation has one term, this is fairly obvious. It is equally true if it has none; for it is not difficult to prove that, if the ultimate oscillation is null, the boundary of the ultimate section is the same as that of the ultimate upper section, and may be defined as the limit of the function for approaches from below. But if the ultimate oscillation has many terms, there is no definite limit to the function for approaches from below. In this case we can take the lower and upper boundaries of the ultimate oscillation (i.e. the lower boundary of the ultimate upper section and the upper boundary of the ultimate section) as the lower and upper limits of its "ultimate" values for approaches from below. Similarly we obtain lower and upper limits of the "ultimate" values for approaches from above. Thus we have, in the general case, four limits to a function for approaches to a given argument. The limit for a given argument only exists when all these four are equal, and is then their common value. If it is also the value for the argument , the function is continuous for this argument. This may be taken as defining continuity: it is equivalent to our former definition.
We can define the limit of a function for a given argument (if it exists) without passing through the ultimate oscillation and the four limits of the general case. The definition proceeds, in that case, just as the earlier definition of continuity proceeded. Let us define the limit for approaches from below. If there is to be a definite limit for approaches to from below, it is necessary and sufficient that, given any small number , two values for arguments sufficiently near to (but both less than ) will differ by less than ; i.e. if is sufficiently small, and our arguments both lie between and ( excluded), then the difference between the values for these arguments will be less than . This is to hold for any , however small; in that case the function has a limit for approaches from below. Similarly we define the case when there is a limit for approaches from above. These two limits, even when both exist, need not be identical; and if they are identical, they still need not be identical with the value for the argument . It is only in this last case that we call the function continuous for the argument .
A function is called "continuous" (without qualification) when it is continuous for every argument.
Another slightly different method of reaching the definition of continuity is the following:—
Let us say that a function "ultimately converges into a class " if there is some real number such that, for this argument and all arguments greater than this, the value of the function is a member of the class . Similarly we shall say that a function "converges into as the argument approaches from below" if there is some argument less than such that throughout the interval from (included) to (excluded) the function has values which are members of . We may now say that a function is continuous for the argument , for which it has the value , if it satisfies four conditions, namely:—
(1) Given any real number less than , the function converges into the successors of this number as the argument approaches from below;
(2) Given any real number greater than , the function converges into the predecessors of this number as the argument approaches from below;
(3) and (4) Similar conditions for approaches to from above.
The advantages of this form of definition is that it analyses the conditions of continuity into four, derived from considering arguments and values respectively greater or less than the argument and value for which continuity is to be defined.
We may now generalise our definitions so as to apply to series which are not numerical or known to be numerically measurable. The case of motion is a convenient one to bear in mind. There is a story by H. G. Wells which will illustrate, from the case of motion, the difference between the limit of a function for a given argument and its value for the same argument. The hero of the story, who possessed, without his knowledge, the power of realising his wishes, was being attacked by a policeman, but on ejaculating "Go to——" he found that the policeman disappeared. If was the policeman's position at time , and the moment of the ejaculation, the limit of the policeman's positions as approached to from below would be in contact with the hero, whereas the value for the argument was —. But such occurrences are supposed to be rare in the real world, and it is assumed, though without adequate evidence, that all motions are continuous, i.e. that, given any body, if is its position at time , is a continuous function of . It is the meaning of "continuity" involved in such statements which we now wish to define as simply as possible.
The definitions given for the case of functions where argument and value are real numbers can readily be adapted for more general use.
Let and be two relations, which it is well to imagine serial, though it is not necessary to our definitions that they should be so. Let be a one-many relation whose domain is contained in the field of , while its converse domain is contained in the field of . Then is (in a generalised sense) a function, whose arguments belong to the field of , while its values belong to the field of . Suppose, for example, that we are dealing with a particle moving on a line: let be the time-series, the series of points on our line from left to right, the relation of the position of our particle on the line at time to the time , so that "the of " is its position at time . This illustration may be borne in mind throughout our definitions.
We shall say that the function is continuous for the argument if, given any interval on the -series containing the value of the function for the argument , there is an interval on the -series containing not as an end-point and such that, throughout this interval, the function has values which are members of . (We mean by an "interval" all the terms between any two; i.e. if and are two members of the field of , and has the relation to , we shall mean by the "-interval to " all terms such that has the relation to and has the relation to —together, when so stated, with or themselves.)
We can easily define the "ultimate section" and the "ultimate oscillation." To define the "ultimate section" for approaches to the argument from below, take any argument which precedes (i.e. has the relation to ), take the values of the function for all arguments up to and including , and form the section of defined by these values, i.e. those members of the -series which are earlier than or identical with some of these values. Form all such sections for all 's that precede , and take their common part; this will be the ultimate section. The ultimate upper section and the ultimate oscillation are then defined exactly as in the previous case.
The adaptation of the definition of convergence and the resulting alternative definition of continuity offers no difficulty of any kind.
We say that a function is "ultimately -convergent into " if there is a member of the converse domain of and the field of such that the value of the function for the argument and for any argument to which has the relation is a member of . We say that "-converges into as the argument approaches a given argument " if there is a term having the relation to and belonging to the converse domain of and such that the value of the function for any argument in the -interval from (inclusive) to (exclusive) belongs to .
Of the four conditions that a function must fulfil in order to be continuous for the argument , the first is, putting for the value for the argument :
Given any term having the relation to , -converges into the successors of (with respect to ) as the argument approaches from below.
The second condition is obtained by replacing by its converse; the third and fourth are obtained from the first and second by replacing by its converse.
There is thus nothing, in the notions of the limit of a function or the continuity of a function, that essentially involves number. Both can be defined generally, and many propositions about them can be proved for any two series (one being the argument-series and the other the value-series). It will be seen that the definitions do not involve infinitesimals. They involve infinite classes of intervals, growing smaller without any limit short of zero, but they do not involve any intervals that are not finite. This is analogous to the fact that if a line an inch long be halved, then halved again, and so on indefinitely, we never reach infinitesimals in this way: after bisections, the length of our bit is of an inch; and this is finite whatever finite number may be. The process of successive bisection does not lead to divisions whose ordinal number is infinite, since it is essentially a one-by-one process. Thus infinitesimals are not to be reached in this way. Confusions on such topics have had much to do with the difficulties which have been found in the discussion of infinity and continuity.