10. Discontinuity of Functions.—The discontinuities of a function of one variable, defined in an interval with the possible exception of isolated points, may be classified as follows:
(1) The function may become infinite, or tend to become infinite, at a point.
(2) The function may be undefined at a point.
(3) The function may have a limit on the left and a limit on the right at the same point; these may be different from each other, and at least one of them must be different from the value of the function at the point.
(4) The function may have no limit at a point, or no limit on the left, or no limit on the right, at a point.
In case a function ƒ(x), defined as above, has no limit at a point a, there are four limiting values which come into consideration. Whatever positive number h we take, the values of the function at points between a and a + h (a excluded) have a superior limit (or a greatest value), and an inferior limit (or a least value); further, as h decreases, the former never increases and the latter never decreases; accordingly each of them tends to a limit. We have in this way two limits on the right—the inferior limit of the superior limits in diminishing neighbourhoods, and the superior limit of the inferior limits in diminishing neighbourhoods. These are denoted by ƒ(a + 0) and ƒ(a + 0), and they are called the “limits of indefiniteness” on the right. Similar limits on the left are denoted by ƒ(a − 0) and ƒ(a − 0). Unless ƒ(x) becomes, or tends to become, infinite at a, all these must exist, any two of them may be equal, and at least one of them must be different from ƒ(a), if ƒ(a) exists. If the first two are equal there is a limit on the right denoted by ƒ(a + 0); if the second two are equal, there is a limit on the left denoted by ƒ(a − 0). In case the function becomes, or tends to become, infinite at a, one or more of these limits is infinite in the sense explained in § 7; and now it is to be noted that, e.g. the superior limit of the inferior limits in diminishing neighbourhoods on the right of a may be negatively infinite; this happens if, after any number N, however great, has been specified, it is possible to find a positive number h, so that all the values of the function in the interval between a and a + h (a excluded) are less than −N; in such a case ƒ(x) tends to become negatively infinite when x decreases towards a; other modes of tending to infinite limits may be described in similar terms.
11. Oscillation of Functions.—The difference between the greatest and least of the numbers ƒ(a), ƒ(a + 0), ƒ(a + 0), ƒ(a − 0), ƒ(a − 0), when they are all finite, is called the “oscillation” or “fluctuation” of the function ƒ(x) at the point a. This difference is the limit for h = 0 of the difference between the superior and inferior limits of the values of the function at points in the interval between a − h and a + h. The corresponding difference for points in a finite interval is called the “oscillation of the function in the interval.” When any of the four limits of indefiniteness is infinite the oscillation is infinite in the sense explained in § 7.
For the further classification of functions we divide the domain of the argument into partial intervals by means of points between the end-points. Suppose that the domain is the interval between a and b. Let intermediate points x1, x2 ... xn−1, be taken so that b > xn−1 > xn−2 ... > x1 > a. We may devise a rule by which, as n increases indefinitely, all the differences b − xn−1, xn−1 − xn−2, ... x1 − a tend to zero as a limit. The interval is then said to be divided into “indefinitely small partial intervals.”
A function defined in an interval with the possible exception of isolated points may be such that the interval can be divided into a set of finite partial intervals within each of which the function is monotonous (§ 8). When this is the case the sum of the oscillations of the function in those partial intervals is finite, provided the function does not tend to become infinite. Further, in such a case the sum of the oscillations will remain below a fixed number for any mode of dividing the interval into indefinitely small partial intervals. A class of functions may be defined by the condition that the sum of the oscillations has this property, and such functions are said to have “restricted oscillation.” Sometimes the phrase “limited fluctuation” is used. It can be proved that any function with restricted oscillation is capable of being expressed as the sum of two monotonous functions, of which one never increases and the other never diminishes throughout the interval. Such a function has a limit on the right and a limit on the left at every point of the interval. This class of functions includes all those which have a finite number of maxima and minima in a finite-interval, and some which have an infinite number. It is to be noted that the class does not include all continuous functions.
12. Differentiable Function.—The idea of the differentiation of a continuous function is that of a process for measuring the rate of growth; the increment of the function is compared with the increment of the variable. If ƒ(x) is defined in an interval containing the point a, and a − k and a + k are points of the interval, the expression