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.