Part 21

7. _Superior and Inferior Limits; Infinities._--The value of a function at every point in the domain of its argument is finite, since, by definition, the value can be assigned, but this does not necessarily imply that there is a number N which exceeds all the values (or is less than all the values). It may happen that, however great a number N we take, there are among the values of the function numbers which exceed N (or are less than -N).

If a number can be found which is greater than every value of the function, then either ([alpha]) there is one value of the function which exceeds all the others, or ([beta]) there is a number S which exceeds every value of the function but is such that, however small a positive number [epsilon] we take, there are values of the function which exceed S -[epsilon]. In the case ([alpha]) the function has a greatest value; in case ([beta]) the function has a "superior limit" S, and then there must be a point a which has the property that there are points of the domain of the argument, in the neighbourhood of a for any h, at which the values of the function differ from S by less than [epsilon]. Thus S is the limit of the function at a, either for the domain of the argument or for some more restricted domain. If a is in the domain of the argument, and if, after omission of a, there is a superior limit S which is in this way the limit of the function at a, if further [f](a) = S, then S is the greatest value of the function: in this case the greatest value is a limit (at any rate for a restricted domain) which is attained; it may be called a "superior limit which is attained." In like manner we may have a "smallest value" or an "inferior limit," and a smallest value may be an "inferior limit which is attained."

All that has been said here may be adapted to the description of greatest values, superior limits, &c., of a function in a restricted domain contained in the domain of the argument. In particular, the domain of the argument may contain an interval; and therein the function may have a superior limit, or an inferior limit, which is attained. Such a limit is a _maximum_ value or a _minimum_ value of the function.

Again, if, after any number N, however great, has been specified, it is possible to find points of the domain of the argument at which the value of the function exceeds N, the values of the function are said to have an "infinite superior limit," and then there must be a point a which has the property that there are points of the domain, in the neighbourhood of a for any h, at which the value of the function exceeds N. If the point a is in the domain of the argument the function is said to "tend to become infinite" at a; it has of course a finite value at a. If the point a is not in the domain of the argument the function is said to "become infinite" at a; it has of course no value at a. In like manner we may have a (negatively) infinite inferior limit. Again, after any number N, however great, has been specified and a number h found, so that all the values of the function, at points in the neighbourhood of a for h, exceed N in absolute value, all these values may have the same sign; the function is then said to become, or to tend to become, "determinately (positively or negatively) infinite"; otherwise it is said to become or to tend to become, "indeterminately infinite."

All the infinities that occur in the theory of functions are of the nature of variable finite numbers, with the single exception of the infinity of an infinite aggregate. The latter is described as an "actual infinity," the former as "improper infinities." There is no "actual infinitely small" corresponding to the actual infinity. The only "infinitely small" is zero. All "infinite values" are of the nature of superior and inferior limits which are not attained.

8. _Increasing and Decreasing Functions._--A function [f](x) of one variable x, defined in the interval between a and b, is "increasing throughout the interval" if, whenever x and x' are two numbers in the interval and x' > x, then [f](x') > [f](x); the function "never decreases throughout the interval" if, x' and x being as before, [f](x') > [f](x). Similarly for decreasing functions, and for functions which never increase throughout an interval. A function which either never increases or never diminishes throughout an interval is said to be "monotonous throughout" the interval. If we take in the above definition b > a, the definition may apply to a function under the restriction that x' is not b and x is not a; such a function is "monotonous within" the interval. In this case we have the theorem that the function (if it never decreases) has a limit on the left at b and a limit on the right at a, and these are the superior and inferior limits of its values at all points within the interval (the ends excluded); the like holds _mutatis mutandis_ if the function never increases. If the function is monotonous throughout the interval, [f](b) is the greatest (or least) value of [f](x) in the interval; and if [f](b) is the limit of [f](x) on the left at b, such a greatest (or least) value is an example of a superior (or inferior) limit which is attained. In these cases the function tends continually to its limit.

These theorems and definitions can be extended, with obvious modifications, to the cases of a domain which is not an interval, or extends to infinite values. By means of them we arrive at sufficient, but not necessary, criteria for the existence of a limit; and these are frequently easier to apply than the general principle of convergence to a limit (S 6), of which principle they are particular cases. For example, the function represented by x log (1/x) continually 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 [f](x) of one variable x is said to be continuous at a point a if (1) [f](x) is defined in an interval containing a; (2) [f](x) has a limit at a; (3) [f](a) is equal to this limit. The limit in question must be a limit for continuous variation, not for a restricted domain. If [f](x) has a limit on the left at a and [f](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 [epsilon], 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 [epsilon]. 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 [epsilon] as in (1) may be the same at all points of the interval, or, in other words, that the numbers h which correspond to [epsilon] 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 [oo] (or at -[oo]).

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 [f](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 /{[f](a + o)} and {[f](a + 0)}/, and they are called the "limits of indefiniteness" on the right. Similar limits on the left are denoted by /{[f](a - 0)} and {[f](a - 0)}/. Unless [f](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 [f](a), if [f](a) exists. If the first two are equal there is a limit on the right denoted by [f](a + 0); if the second two are equal, there is a limit on the left denoted by [f](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 S 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 [f](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 [f](a), /{[f](a + 0)}, {[f](a + 0)}/, /{[f](a - 0)}, {[f](a - 0)}/, when they are all finite, is called the "oscillation" or "fluctuation" of the function [f](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 S 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 ... x{n - 1}_, be taken so that b > X_(n - 1) > x_(n - 2) ... > X1 > a_. We may devise a rule by which, as n increases indefinitely, all the differences b - x_(n - 1), x_(n - 1) - x_(n - 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 (S 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 _[f](x)_ is defined in an interval containing the point a, and _a - k_ and _a + k_ are points of the interval, the expression

[f](a + h) - [f](a) ------------------- (1) h

represents a function of h, which we may call [phi](h), defined at all points of an interval for h between -k and k except the point 0. Thus the four limits /[phi](+0), [phi](+0)/, /[phi](-0), [phi](-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 [f](x) at the point a; when the last two are equal either of them is the "regressive differential coefficient" of [f](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 [f](x) at a, or the "first derived function" of [f](x) at a. It is denoted by d[f](x) / dx or by [f]'(x). In this case [phi](h) has a definite limit at h = 0, or is determinately infinite at h = 0 (S 7). The four limits here in question are called, after Dini, the "four derivates" of [f](x) at a. In accordance with the notation for derived functions they may be denoted by

---------- ---------- [f]' + (a), [f]' + (a), [f]' - (a), f' - (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 Ampere 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 S 24). The most important theorem in regard to differentiable functions is the "theorem of intermediate value." (See INFINITESIMAL CALCULUS.)

13. _Analytic Function._--If [f](x) and its first n differential coefficients, denoted by[f]'(x), [f]''(x), ... [f](^n)(x), are continuous in the interval between a and a + h, then

h^2 [f](a + h) = [f](a) + h[f]'(a) + --- f''(a) + ... 2!

h^(n - 1) + --------- [f]^(n - 1)(a) + R_n, (n - 1)!

where R_n may have various forms, some of which are given in the article INFINITESIMAL CALCULUS. This result is known as "Taylor's theorem."

When Taylor's theorem leads to a representation of the function by means of an infinite series, the function is said to be "analytic" (cf. S 21).

14. _Ordinary Function._--The idea of a curve representing a continuous function in an interval is that of a line which has the following properties: (1) the co-ordinates of a point of the curve are a value x of the argument and the corresponding value y of the function; (2) at every point the curve has a definite tangent; (3) the interval can be divided into a finite number of partial intervals within each of which the function is monotonous; (4) the property of monotony within partial intervals is retained after interchange of the axes of co-ordinates x and y. According to condition (2) y is a continuous and differentiable function of x, but this condition does not include conditions (3) and (4): there are continuous partially monotonous functions which are not differentiable, there are continuous differentiable functions which are not monotonous in any interval however small; and there are continuous, differentiable and monotonous functions which do not satisfy condition (4) (cf. S 24). A function which can be represented by a curve, in the sense explained above, is said to be "ordinary," and the curve is the graph of the function (S2). All analytic functions are ordinary, but not all ordinary functions are analytic.

15. _Integrable Function._--The idea of integration is twofold. We may seek the function which has a given function as its differential coefficient, or we may generalize the question of finding the area of a curve. The first inquiry leads directly to the indefinite integral, the second directly to the definite integral. Following the second method we define "the definite integral of the function [f](x) through the interval between a and b" to be the limit of the sum

_n \ [f](x'_r)(x_r - x_(r - 1)) /_ 1

when the interval is divided into ultimately indefinitely small partial intervals by points x1, x2, ... x_(n - 1). Here x'_r denotes any point in the rth partial interval, x0 is put for a, and x_n for b. It can be shown that the limit in question is finite and independent of the mode of division into partial intervals, and of the choice of the points such as x'_r, provided (1) the function is defined for all points of the interval, and does not tend to become infinite at any of them; (2) for any one mode of division of the interval into ultimately indefinitely small partial intervals, the sum of the products of the oscillation of the function in each partial interval and the difference of the end-values of that partial interval has limit zero when n is increased indefinitely. When these conditions are satisfied the function is said to be "integrable" in the interval. The numbers a and b which limit the interval are usually called the "lower and upper limits." We shall call them the "nearer and further end-values." The above definition of integration was introduced by Riemann in his memoir on trigonometric series (1854). A still more general definition has been given by Lebesgue. As the more general definition cannot be made intelligible without the introduction of some rather recondite notions belonging to the theory of aggregates, we shall, in what follows, adhere to Riemann's definition.

We have the following theorems:--

1. Any continuous function is integrable.

2. Any function with restricted oscillation is integrable.

3. A discontinuous function is integrable if it does not tend to become infinite, and if the points at which the oscillation of the function exceeds a given number [sigma], however small, can be enclosed in partial intervals the sum of whose breadths can be diminished indefinitely.

These partial intervals must be a set chosen out of some complete set obtained by the process used in the definition of integration.

4. The sum or product of two integrable functions is integrable.

As regards integrable functions we have the following theorems:

1. If S and I are the superior and inferior limits (or greatest and least values) of [f](x) in the interval between a and b, [int] [a to b] [f](x)dx is intermediate between S(b - a) and I(b - a).

2. The integral is a continuous function of each of the end-values.

3. If the further end-value b is variable, and if [int] [a to x] [f](x)_dx_ = F(x), then if [f](x) is continuous at b, F(x) is differentiable at b, and F'(b) = [f](b).

4. In case [f](x) is continuous throughout the interval F(x) is continuous and differentiable throughout the interval, and F'(x) = [f](x) throughout the interval.

5. In case [f]'(x) is continuous throughout the interval between a and b, _ / b | [f]'(x)dx = [f](b) - [f](a). _/a

6. In case [f](x) is discontinuous at one or more points of the interval between a and b, in which it is integrable, _ / x | [f](x)dx _/a

is a function of x, of which the four derivates at any point of the interval are equal to the limits of indefiniteness of [f](x) at the point.

7. It may be that there exist functions which are differentiable throughout an interval in which their differential coefficients are not integrable; if, however, F(x) is a function whose differential coefficient, F'(x), is integrable in an interval, then _ / x F(x) = | F'(x)dx + const., _/a

where a is a fixed point, and x a variable point, of the interval. Similarly, if any one of the four derivates of a function is integrable in an interval, all are integrable, and the integral of either differs from the original function by a constant only.

The theorems (4), (6), (7) show that there is some discrepancy between the indefinite integral considered as the function which has a given function as its differential coefficient, and as a definite integral with a variable end-value.