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.