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 [Pg 112] 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:—