diminishes when 1/e > x > 0 and x diminishes towards zero, and it never becomes negative. It therefore has a limit on the right at x = 0. This limit is zero. The function represented by x sin (1/x) does not continually diminish towards zero as x diminishes towards zero, but is sometimes greater than zero and sometimes less than zero in any neighbourhood of x = 0, however small. Nevertheless, the function has the limit zero at x = 0.
9. Continuity of Functions.—A function ƒ(x) of one variable x is said to be continuous at a point a if (1) ƒ(x) is defined in an interval containing a; (2) ƒ(x) has a limit at a; (3) ƒ(a) is equal to this limit. The limit in question must be a limit for continuous variation, not for a restricted domain. If ƒ(x) has a limit on the left at a and ƒ(a) is equal to this limit, the function may be said to be “continuous to the left” at a; similarly the function may be “continuous to the right” at a.
A function is said to be “continuous throughout an interval” when it is continuous at every point of the interval. This implies continuity to the right at the smaller end-value and continuity to the left at the greater end-value. When these conditions at the ends are not satisfied the function is said to be continuous “within” the interval. By a “continuous function” of one variable we always mean a function which is continuous throughout an interval.
The principal properties of a continuous function are:
1. The function is practically constant throughout sufficiently small intervals. This means that, after any point a of the interval has been chosen, and any positive number ε, however small, has been specified, it is possible to find a number h, so that the difference between any two values of the function in the interval between a − h and a + h is less than ε. There is an obvious modification if a is an end-point of the interval.
2. The continuity of the function is “uniform.” This means that the number h which corresponds to any ε as in (1) may be the same at all points of the interval, or, in other words, that the numbers h which correspond to ε for different values of a have a positive inferior limit.
3. The function has a greatest value and a least value in the interval, and these are superior and inferior limits which are attained.
4. There is at least one point of the interval at which the function takes any value between its greatest and least values in the interval.
5. If the interval is unlimited towards the right (or towards the left), the function has a limit at ∞ (or at −∞).
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
| ƒ(a + h) − ƒ(a)h | (1) |
represents a function of h, which we may call φ(h), defined at all points of an interval for h between −k and k except the point 0. Thus the four limits φ(+0), φ(+0), φ(−0), φ(−0) exist, and two or more of them may be equal. When the first two are equal either of them is the “progressive differential coefficient” of ƒ(x) at the point a; when the last two are equal either of them is the “regressive differential coefficient” of ƒ(x) at a; when all four are equal the function is said to be “differentiable” at a, and either of them is the “differential coefficient” of ƒ(x) at a, or the “first derived function” of ƒ(x) at a. It is denoted by dƒ(x)dx or by ƒ′(x). In this case φ(h) has a definite limit at h = 0, or is determinately infinite at h = 0 (§ 7). The four limits here in question are called, after Dini, the “four derivates” of ƒ(x) at a. In accordance with the notation for derived functions they may be denoted by
ƒ′ + (a), ƒ′ + (a), ƒ′ − (a), ƒ′ − (a).
A function which has a finite differential coefficient at all points of an interval is continuous throughout the interval, but if the differential coefficient becomes infinite at a point of the interval the function may or may not be continuous throughout the interval; on the other hand a function may be continuous without being differentiable. This result, comparable in importance, from the point of view of the general theory of functions, with the discovery of Fourier’s theorem, is due to G. F. B. Riemann; but the failure of an attempt made by Ampère to prove that every continuous function must be differentiable may be regarded as the first step in the theory. Examples of analytical expressions which represent continuous functions that are not differentiable have been given by Riemann, Weierstrass, Darboux and Dini (see § 24). The most important theorem in regard to differentiable functions is the “theorem of intermediate value.” (See Infinitesimal Calculus.)
