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 [Pg 110] 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. [Pg 111] 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 .