Introduction to Mathematical Philosophy

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. [Pg 113] 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 [Pg 114] 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 : [Pg 115] 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. [Pg 116]