For the specification of any particular function two things are requisite: (1) a statement of the values of the variable, or of the aggregate of points, to which values of the function are to be made to correspond,i.e.of the “domain of the argument”; (2) a rule for assigning the value or values of the function that correspond to any point in this domain. We may refer to the second of these two essentials as “the rule of calculation.” The relation of functions to analytical expressions may then be stated in the form that the rule of calculation is: “Give the function the value of the expression at any point at which the expression has a determinate value,” or again more generally, “Give the function the value of the expression at all points of a definite aggregate included in the domain of the argument.” The former of these is the rule of those among the earlier analysts who regarded an analytical expression and a function as the same thing, and their usage may be retained without causing confusion and with the advantage of brevity, the analytical expression serving to specify the domain of the argument as well as the rule of calculation,e.g.we may speak of “the function 1/x.” This function is defined by the analytical expression 1/x at all points except the point x = 0. But in complicated cases separate statements of the domain of the argument and the rule of calculation cannot be dispensed with. In general, when the rule of calculation is determined as above by an analytical expression at any aggregate of points, the function is said to be “represented” by the expression at those points.When the rule of calculation assigns a single definite value for a function at each point in the domain of the argument the function is “uniform” or “one-valued.” In what follows it is to be understood that all the functions considered are one-valued, and the valuesassigned by the rule of calculation real. In the most important cases the domain of the argument of a function of one variable is an interval, with the possible exception of isolated points.
For the specification of any particular function two things are requisite: (1) a statement of the values of the variable, or of the aggregate of points, to which values of the function are to be made to correspond,i.e.of the “domain of the argument”; (2) a rule for assigning the value or values of the function that correspond to any point in this domain. We may refer to the second of these two essentials as “the rule of calculation.” The relation of functions to analytical expressions may then be stated in the form that the rule of calculation is: “Give the function the value of the expression at any point at which the expression has a determinate value,” or again more generally, “Give the function the value of the expression at all points of a definite aggregate included in the domain of the argument.” The former of these is the rule of those among the earlier analysts who regarded an analytical expression and a function as the same thing, and their usage may be retained without causing confusion and with the advantage of brevity, the analytical expression serving to specify the domain of the argument as well as the rule of calculation,e.g.we may speak of “the function 1/x.” This function is defined by the analytical expression 1/x at all points except the point x = 0. But in complicated cases separate statements of the domain of the argument and the rule of calculation cannot be dispensed with. In general, when the rule of calculation is determined as above by an analytical expression at any aggregate of points, the function is said to be “represented” by the expression at those points.
When the rule of calculation assigns a single definite value for a function at each point in the domain of the argument the function is “uniform” or “one-valued.” In what follows it is to be understood that all the functions considered are one-valued, and the valuesassigned by the rule of calculation real. In the most important cases the domain of the argument of a function of one variable is an interval, with the possible exception of isolated points.
6.Limits.—Let ƒ(x) be a function of a variable number x; and let a be a point such that there are points of the domain of the argument x in the neighbourhood of a for any number h, however small. If there is a number L which has the property that, after any positive number ε, however small, has been specified, it is possible to find a positive number h, so that |L − ƒ(x)| < ε for all points x of the domain (other than a) for which |x − a| < h, then L is the “limit of ƒ(x) at the point a.” The condition for the existence of L is that, after the positive number ε has been specified, it must be possible to find a positive number h, so that |ƒ(x′) − ƒ(x)| < ε for all points x and x′ of the domain (other than a) for which |x − a| < h and |x′ − a| < h.
It is a fundamental theorem that, when this condition is satisfied, there exists a perfectly definite number L which is the limit of ƒ(x) at the point a as defined above. The limit of ƒ(x) at the point a is denoted by Ltx=aƒ(x), or by limx=aƒ(x).
If ƒ(x) is a function of one variable x in a domain which extends to infinite values, and if, after ε has been specified, it is possible to find a number N, so that |ƒ(x′) − ƒ(x)| < ε for all values of x and x′ which are in the domain and exceed N, then there is a number L which has the property that |ƒ(x) − L| < ε for all such values of x. In this case ƒ(x) has a limit L at x = ∞. In like manner ƒ(x) may have a limit at x = −∞. This statement includes the case where the domain of the argument consists exclusively of positive integers. The values of the function then form a “sequence,” u1, u2, ... un, ..., and this sequence can have a limit at n = ∞.The principle common to the above definitions and theorems is called, after P. du Bois Reymond, “the general principle of convergence to a limit.”It must be understood that the phrase “x = ∞” does not mean that x takes some particular value which is infinite. There is no such value. The phrase always refers to a limiting process in which, as the process is carried out, the variable number x increases without limit: it may, as in the above example of a sequence, increase by taking successively the values of all the integral numbers; in other cases it may increase by taking the values that belong to any domain which “extends to infinite values.”A very important type of limits is furnished byinfinite series. When a sequence of numbers u1, u2, ... un, ... is given, we may form a new sequence s1, s2, ... sn, ... from it by the rules s1= u1, s2= u1+ u2, ... sn= u1+ u2+ ... + unor by the equivalent rules s1= u, sn− sn−1= un(n = 2, 3, ...). If the new sequence has a limit at n = ∞, this limit is called the “sum of the infinite series” u1+ u2+ ..., and the series is said to be “convergent” (seeSeries).A function which has not a limit at a point a may be such that, if a certain aggregate of points is chosen out of the domain of the argument, and the points x in the neighbourhood of a are restricted to belong to this aggregate, then the function has a limit at a. For example, sin(1/x) has limit zero at 0 if x is restricted to the aggregate 1/π, 1/2π, ... 1/nπ, ... or to the aggregate 1/2π, 2/5π, ... n/(n2+ 1)π, ..., but if x takes all values in the neighbourhood of 0, sin (1/x) has not a limit at 0. Again, there may be a limit at a if the points x in the neighbourhood of a are restricted by the condition that x − a is positive; then we have a “limit on the right” at a; similarly we may have a “limit on the left” at a point. Any such limit is described as a “limit for a restricted domain.” The limits on the left and on the right are denoted by ƒ(a − 0) and ƒ(a + 0).The limit L of ƒ(x) at a stands in no necessary relation to the value of ƒ(x) at a. If the point a is in the domain of the argument, the value of ƒ(x) at a is assigned by the rule of calculation, and may be different from L. In case ƒ(a) = L the limit is said to be “attained.” If the point a is not in the domain of the argument, there is no value for ƒ(x) at a. In the case where ƒ(x) is defined for all points in an interval containing a, except the point a, and has a limit L at a, we may arbitrarily annex the point a to the domain of the argument and assign to ƒ(a) the value L; the function may then be said to be “extrinsically defined.” The so-called “indeterminate forms” (seeInfinitesimal Calculus) are examples.
If ƒ(x) is a function of one variable x in a domain which extends to infinite values, and if, after ε has been specified, it is possible to find a number N, so that |ƒ(x′) − ƒ(x)| < ε for all values of x and x′ which are in the domain and exceed N, then there is a number L which has the property that |ƒ(x) − L| < ε for all such values of x. In this case ƒ(x) has a limit L at x = ∞. In like manner ƒ(x) may have a limit at x = −∞. This statement includes the case where the domain of the argument consists exclusively of positive integers. The values of the function then form a “sequence,” u1, u2, ... un, ..., and this sequence can have a limit at n = ∞.
The principle common to the above definitions and theorems is called, after P. du Bois Reymond, “the general principle of convergence to a limit.”
It must be understood that the phrase “x = ∞” does not mean that x takes some particular value which is infinite. There is no such value. The phrase always refers to a limiting process in which, as the process is carried out, the variable number x increases without limit: it may, as in the above example of a sequence, increase by taking successively the values of all the integral numbers; in other cases it may increase by taking the values that belong to any domain which “extends to infinite values.”
A very important type of limits is furnished byinfinite series. When a sequence of numbers u1, u2, ... un, ... is given, we may form a new sequence s1, s2, ... sn, ... from it by the rules s1= u1, s2= u1+ u2, ... sn= u1+ u2+ ... + unor by the equivalent rules s1= u, sn− sn−1= un(n = 2, 3, ...). If the new sequence has a limit at n = ∞, this limit is called the “sum of the infinite series” u1+ u2+ ..., and the series is said to be “convergent” (seeSeries).
A function which has not a limit at a point a may be such that, if a certain aggregate of points is chosen out of the domain of the argument, and the points x in the neighbourhood of a are restricted to belong to this aggregate, then the function has a limit at a. For example, sin(1/x) has limit zero at 0 if x is restricted to the aggregate 1/π, 1/2π, ... 1/nπ, ... or to the aggregate 1/2π, 2/5π, ... n/(n2+ 1)π, ..., but if x takes all values in the neighbourhood of 0, sin (1/x) has not a limit at 0. Again, there may be a limit at a if the points x in the neighbourhood of a are restricted by the condition that x − a is positive; then we have a “limit on the right” at a; similarly we may have a “limit on the left” at a point. Any such limit is described as a “limit for a restricted domain.” The limits on the left and on the right are denoted by ƒ(a − 0) and ƒ(a + 0).
The limit L of ƒ(x) at a stands in no necessary relation to the value of ƒ(x) at a. If the point a is in the domain of the argument, the value of ƒ(x) at a is assigned by the rule of calculation, and may be different from L. In case ƒ(a) = L the limit is said to be “attained.” If the point a is not in the domain of the argument, there is no value for ƒ(x) at a. In the case where ƒ(x) is defined for all points in an interval containing a, except the point a, and has a limit L at a, we may arbitrarily annex the point a to the domain of the argument and assign to ƒ(a) the value L; the function may then be said to be “extrinsically defined.” The so-called “indeterminate forms” (seeInfinitesimal Calculus) are examples.
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 (α) there is one value of the function which exceeds all the others, or (β) there is a number S which exceeds every value of the function but is such that, however small a positive number ε we take, there are values of the function which exceed S − ε. In the case (α) the function has a greatest value; in case (β) 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 ε. 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 ƒ(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 amaximumvalue or aminimumvalue 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.
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 amaximumvalue or aminimumvalue 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 ƒ(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 ƒ(x′) > ƒ(x); the function “never decreases throughout the interval” if, x′ and x being as before, ƒ(x′) > ƒ(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 holdsmutatis mutandisif the function never increases. If the function is monotonous throughout the interval, ƒ(b) is the greatest (or least) value of ƒ(x) in the interval; and if ƒ(b) is the limit of ƒ(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 (§ 6), of which principle they are particular cases. For example, the function represented by x log (1/x) continuallydiminishes 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.
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 (§ 6), of which principle they are particular cases. For example, the function represented by x log (1/x) continuallydiminishes 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 −∞).
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 byƒ(a + 0)andƒ(a + 0), and they are called the “limits of indefiniteness” on the right. Similar limits on the left are denoted byƒ(a − 0)andƒ(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.
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 byƒ(a + 0)andƒ(a + 0), and they are called the “limits of indefiniteness” on the right. Similar limits on the left are denoted byƒ(a − 0)andƒ(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.
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
(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 limitsφ(+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.” (SeeInfinitesimal Calculus.)
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.” (SeeInfinitesimal Calculus.)
13.Analytic Function.—If ƒ(x) and its first n differential coefficients, denoted byƒ′(x), ƒ″(x), ... ƒ(n) (x), are continuous in the interval between a and a + h, then
where Rnmay have various forms, some of which are given in the articleInfinitesimal Calculus. This result is known as “Taylor’s theorem.”
WhenTaylor'stheorem leads to a representation of the function by means of an infinite series, the function is said to be “analytic” (cf. § 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. § 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 (§2). 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 ƒ(x) through the interval between a and b” to be the limit of the sum
Σn1ƒ(x′r) (xr− xr−1)
when the interval is divided into ultimately indefinitely small partial intervals by points x1, x2, ... xn−1. Here x′rdenotes any point in the rth partial interval, x0is put for a, and xnfor 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 σ, 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 ƒ(x) in the interval between a and b,∫baƒ(x)dxis 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∫xaƒ(x)dx= F(x), then if ƒ(x) is continuous at b, F(x) is differentiable at b, and F′(b) = ƒ(b).4. In case ƒ(x) is continuous throughout the interval F(x) is continuous and differentiable throughout the interval, and F′(x) = ƒ(x) throughout the interval.5. In case ƒ′(x) is continuous throughout the interval between a and b,∫baƒ′(x)dx= ƒ(b) − ƒ(a).6. In case ƒ(x) is discontinuous at one or more points of the interval between a and b, in which it is integrable,∫xaƒ(x)dxis a function of x, of which the four derivates at any point of the interval are equal to the limits of indefiniteness of ƒ(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, thenF(x) =∫xaF′(x)dx+ const.,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.We have also two theorems concerning the integral of the product of two integrable functions ƒ(x) and φ(x); these are known as “the first and second theorems of the mean.” The first theorem of the mean is that, if φ(x) is one-signed throughout the interval between a and b, there is a number M intermediate between the superior and inferior limits, or greatest and least values, of ƒ(x) in the interval, which has the property expressed by the equationM∫baφ(x)dx =∫baƒ(x)φ(x)dxThe second theorem of the mean is that, if ƒ(x) is monotonous throughout the interval, there is a number ξ between a and b which has the property expressed by the equation∫baƒ(x) φ(x)dx = ƒ(a)∫ξaφ(x)dx+ ƒ(b)∫bξφ(x)dx.(SeeFourier’s Series.)
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 σ, 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 ƒ(x) in the interval between a and b,∫baƒ(x)dxis 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∫xaƒ(x)dx= F(x), then if ƒ(x) is continuous at b, F(x) is differentiable at b, and F′(b) = ƒ(b).
4. In case ƒ(x) is continuous throughout the interval F(x) is continuous and differentiable throughout the interval, and F′(x) = ƒ(x) throughout the interval.
5. In case ƒ′(x) is continuous throughout the interval between a and b,
∫baƒ′(x)dx= ƒ(b) − ƒ(a).
6. In case ƒ(x) is discontinuous at one or more points of the interval between a and b, in which it is integrable,
∫xaƒ(x)dx
is a function of x, of which the four derivates at any point of the interval are equal to the limits of indefiniteness of ƒ(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
F(x) =∫xaF′(x)dx+ const.,
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.
We have also two theorems concerning the integral of the product of two integrable functions ƒ(x) and φ(x); these are known as “the first and second theorems of the mean.” The first theorem of the mean is that, if φ(x) is one-signed throughout the interval between a and b, there is a number M intermediate between the superior and inferior limits, or greatest and least values, of ƒ(x) in the interval, which has the property expressed by the equation
M∫baφ(x)dx =∫baƒ(x)φ(x)dx
The second theorem of the mean is that, if ƒ(x) is monotonous throughout the interval, there is a number ξ between a and b which has the property expressed by the equation
∫baƒ(x) φ(x)dx = ƒ(a)∫ξaφ(x)dx+ ƒ(b)∫bξφ(x)dx.
(SeeFourier’s Series.)
16.Improper Definite Integrals.—We may extend the idea of integration to cases of functions which are not defined at some point, or which tend to become infinite in the neighbourhood of some point, and to cases where the domain of the argument extends to infinite values. If c is a point in the interval between a and b at which ƒ(x) is not defined, we impose a restriction on the points x′rof the definition: none of them is to be the point c. This comes to the same thing as defining∫baƒ(x)dxto be
Ltε=0∫c−εaƒ(x)dx + Ltε′=0∫bc+ε′ƒ(x)dx,
(1)
where, to fix ideas, b is taken > a, and ε and ε′ are positive. The same definition applies to the case where ƒ(x) becomes infinite, or tends to become infinite, at c, provided both the limits exist. This definition may be otherwise expressed by saying that a partial interval containing the point c is omitted from the interval of integration, and a limit taken by diminishing the breadth of this partial interval indefinitely; in this form it applies to the cases where c is a or b.
Again, when the interval of integration is unlimited to the right, or extends to positively infinite values, we have as a definition
∫∞aƒ(x)dx = Lth=∞∫haƒ(x)dx,
provided this limit exists. Similar definitions apply to
∫−∞aƒ(x)dx, and to∫∞−∞ƒ(x)dx.
All such definite integrals as the above are said to be “improper.” For example,∫∞0sin x / x dx is improper in two ways. It means
Lth=∞Ltε=0∫hεsinx/x dx,
in which the positive number ε is first diminished indefinitely, and the positive number h is afterwards increased indefinitely.
The “theorems of the mean” (§ 15) require modification when the integrals are improper (seeFourier’s Series).
When the improper definite integral of a function which becomes, or tends to become, infinite, exists, the integral is said to be “convergent.” If ƒ(x) tends to become infinite at a point c in the interval between a and b, and the expression (1) does not exist, then the expression∫baƒ(x)dx, which has no value, is called a “divergent integral, “and it may happen that there is a definite value for
Lt{∫c−εaƒ(x) dx +∫bc+ε′ƒ(x) dx}
provided that ε and ε′ are connected by some definite relation, and both, remaining positive, tend to limit zero. The value of the above limit is then called a “principal value” of the divergent integral. Cauchy’s principal value is obtained by making ε′ = ε,i.e.by taking the omitted interval so that the infinity is at its middle point. A divergent integral which has one or more principal values is sometimes described as “semi-convergent.”
17.Domain of a Set of Variables.—The numerical continuum of n dimensions (Cn) is the aggregate that is arrived at by attributing simultaneous values to each of n variables x1, x2, ... xn, these values being any real numbers. The elements of such an aggregate are called “points,” and the numbers x1, x2... xnthe “co-ordinates” of a point. Denoting in general the points (x1, x2, ... xn) and (x′1, x′2... x′n) by x and x′, the sum of the differences |x1− x′1| + |x2− x′2| + ... + |xn− x′n| may be denoted by |x − x′| and called the “difference of the two points.” We can in various ways choose out of the continuum an aggregate of points, which may be an infinite aggregate, and any such aggregate can be the “domain” of a “variable point.” The domain is said to “extend to an infinite distance” if, after any number N, however great, has been specified, it is possible to find in the domain points of which one or more co-ordinates exceed N in absolute value. The “neighbourhood” of a point a for a (positive) number h is the aggregate constituted of all the points x, which are such that the “difference” denoted by |x − a| < h. If an infinite aggregate of points does not extend to an infinite distance, there must be at least one point a, which has the property that the points of the aggregate which are in the neighbourhood of a for any number h, however small, themselves constitute an infinite aggregate, and then the point a is called a “limiting point” of the aggregate; it may or may not be a point of the aggregate. An aggregate of points is “perfect” when all its points are limiting points of it, and all its limiting points are points of it; it is “connected” when, after taking any two points a, b of it, and choosing any positive number ε, however small, a number m and points x′, x″, ... x(m)of the aggregate can be found so that all the differences denoted by |x′ − a|, |x″ − x′|, ... |b − x(m)| are less than ε. A perfect connected aggregate is acontinuum. This is G. Cantor’s definition.
The definition of a continuum in Cnleaves open the question of the number of dimensions of the continuum, and a further explanation is necessary in order to define arithmetically what is meant by a “homogeneous part” Hnof Cn. Such a part would correspond to an interval in C1, or to an area bounded by a simple closed contour in C2; and, besides being perfect and connected, it would have the following properties: (1) There are points of Cn, which are not points of Hn; these form a complementary aggregate H′n. (2) There are points “within” Hn; this means that for any such point there is a neighbourhood consisting exclusively of points of Hn. (3) The points of Hnwhich do not lie “within” Hnare limiting points of H′n; they are not points of H′n, but the neighbourhood of any such point for any number h, however small, contains points within Hnand points of H′n: the aggregate of these points is called the “boundary” of Hn. (4) When any two points a, b within Hnare taken, it is possible to find a number ε and a corresponding number m, and to choose points x′, x″, ... x(m), so that the neighbourhood of a for ε contains x′, and consists exclusively of points within Hn, and similarly for x′ and x″, x″ and x″′, ... x(m)and b. Condition (3) would exclude such an aggregate as that of the points within and upon two circles external to each other and a line joining a point on one to a point on the other, and condition (4) would exclude such an aggregate as that of the points within and upon two circles which touch externally.
The definition of a continuum in Cnleaves open the question of the number of dimensions of the continuum, and a further explanation is necessary in order to define arithmetically what is meant by a “homogeneous part” Hnof Cn. Such a part would correspond to an interval in C1, or to an area bounded by a simple closed contour in C2; and, besides being perfect and connected, it would have the following properties: (1) There are points of Cn, which are not points of Hn; these form a complementary aggregate H′n. (2) There are points “within” Hn; this means that for any such point there is a neighbourhood consisting exclusively of points of Hn. (3) The points of Hnwhich do not lie “within” Hnare limiting points of H′n; they are not points of H′n, but the neighbourhood of any such point for any number h, however small, contains points within Hnand points of H′n: the aggregate of these points is called the “boundary” of Hn. (4) When any two points a, b within Hnare taken, it is possible to find a number ε and a corresponding number m, and to choose points x′, x″, ... x(m), so that the neighbourhood of a for ε contains x′, and consists exclusively of points within Hn, and similarly for x′ and x″, x″ and x″′, ... x(m)and b. Condition (3) would exclude such an aggregate as that of the points within and upon two circles external to each other and a line joining a point on one to a point on the other, and condition (4) would exclude such an aggregate as that of the points within and upon two circles which touch externally.
18. Functions of Several Variables.—A function of several variables differs from a function of one variable in that the argument of the function consists of a set of variables, or is a variable point in a Cnwhen there are n variables. The function is definable by means of the domain of the argument and the rule of calculation. In the most important cases the domain of the argument is a homogeneous part Hnof Cnwith the possible exception of isolated points, and the rule of calculation is that the value of the function in any assigned part of the domain of the argument is that value which is assumed at the point by an assigned analytical expression. The limit of a function at a point a is defined in the same way as in the case of a function of one variable.
We take a positive fraction ε and consider the neighbourhood of a for h, and from this neighbourhood we exclude the point a, and we also exclude any point which is not in the domain of the argument. Then we take x and x′ to be any two of the retained points in the neighbourhood. The function ƒ has a limit at a if for any positive ε, however small, there is a corresponding h which has the property that |ƒ(x′) − ƒ(x)| < ε, whatever points x, x′ in the neighbourhood of a for h we take (a excluded). For example, when there are two variables x1, x2, and both are unrestricted, the domain of the argument is represented by a plane, and the values of the function are correlated with the points of the plane. The function has a limit at a point a, if we can mark out on the plane a region containing the point a within it, and such that the difference of the values of the function which correspond to any two points of the region (neither of the points being a) can be made as small as we please in absolute value by contracting all the linear dimensions of the region sufficiently. When the domain of the argument of a function of n variables extends to an infinite distance, there is a “limit at an infinite distance” if, after any number ε, however small, has been specified, a number N can be found which is such that |ƒ(x′) − ƒ(x)| < ε, for all points x and x′ (of the domain) of which one or more co-ordinates exceed N in absolute value. In the case of functions of several variables great importance attaches to limits for a restricted domain. The definition of such a limit is verbally the same as the corresponding definition in the case of functions of one variable (§ 6). For example, a function of x1and x2may have a limit at (x1= 0, x2= 0) if we first diminish x1without limit, keeping x2constant, and afterwards diminish x2without limit. Expressed in geometrical language, this process amounts to approaching the origin along the axis of x2. The definitions of superior and inferior limits, and of maxima and minima, and the explanations of what is meant by saying that a function of several variables becomes infinite, or tends to become infinite, at a point, are almost identical verbally with the corresponding definitions and explanations in the case of a function of one variable (§ 7). The definition of a continuous function (§ 9) admits of immediate extension; but it is very important to observe that a function of two or more variables may be a continuous function of each of the variables, when the rest are kept constant, without being a continuous function of its argument. For example, a function of x and y may be defined by the conditions that when x = 0 it is zero whatever value y may have, and when x ≠ 0 it has the value of sin {4 tan−1(y/x)}. When y has any particular value this function is a continuous function of x, and, when x has any particular value this function is a continuous function of y; but the function of x and y is discontinuous at (x = 0, y = 0).
We take a positive fraction ε and consider the neighbourhood of a for h, and from this neighbourhood we exclude the point a, and we also exclude any point which is not in the domain of the argument. Then we take x and x′ to be any two of the retained points in the neighbourhood. The function ƒ has a limit at a if for any positive ε, however small, there is a corresponding h which has the property that |ƒ(x′) − ƒ(x)| < ε, whatever points x, x′ in the neighbourhood of a for h we take (a excluded). For example, when there are two variables x1, x2, and both are unrestricted, the domain of the argument is represented by a plane, and the values of the function are correlated with the points of the plane. The function has a limit at a point a, if we can mark out on the plane a region containing the point a within it, and such that the difference of the values of the function which correspond to any two points of the region (neither of the points being a) can be made as small as we please in absolute value by contracting all the linear dimensions of the region sufficiently. When the domain of the argument of a function of n variables extends to an infinite distance, there is a “limit at an infinite distance” if, after any number ε, however small, has been specified, a number N can be found which is such that |ƒ(x′) − ƒ(x)| < ε, for all points x and x′ (of the domain) of which one or more co-ordinates exceed N in absolute value. In the case of functions of several variables great importance attaches to limits for a restricted domain. The definition of such a limit is verbally the same as the corresponding definition in the case of functions of one variable (§ 6). For example, a function of x1and x2may have a limit at (x1= 0, x2= 0) if we first diminish x1without limit, keeping x2constant, and afterwards diminish x2without limit. Expressed in geometrical language, this process amounts to approaching the origin along the axis of x2. The definitions of superior and inferior limits, and of maxima and minima, and the explanations of what is meant by saying that a function of several variables becomes infinite, or tends to become infinite, at a point, are almost identical verbally with the corresponding definitions and explanations in the case of a function of one variable (§ 7). The definition of a continuous function (§ 9) admits of immediate extension; but it is very important to observe that a function of two or more variables may be a continuous function of each of the variables, when the rest are kept constant, without being a continuous function of its argument. For example, a function of x and y may be defined by the conditions that when x = 0 it is zero whatever value y may have, and when x ≠ 0 it has the value of sin {4 tan−1(y/x)}. When y has any particular value this function is a continuous function of x, and, when x has any particular value this function is a continuous function of y; but the function of x and y is discontinuous at (x = 0, y = 0).
19.Differentiation and Integration.—The definition of partial differentiation of a function of several variables presents no difficulty. The most important theorems concerning differentiable functions are the “theorem of the total differential,” the theorem of the interchangeability of the order of partial differentiations, and the extension of Taylor’s theorem (seeInfinitesimal Calculus).
With a view to the establishment of the notion of integration through a domain, we must define the “extent” of the domain. Take first a domain consisting of the point a and all the points x for which |x − a| < ½h, where h is a chosen positive number; the extent of this domain is hn, n being the number of variables; such a domain may be described as “square,” and the number h may be called its “breadth”; it is a homogeneous part of thenumerical continuum of n dimensions, and its boundary consists of all the points for which |x − a| = ½h. Now the points of any domain, which does not extend to an infinite distance, may be assigned to a finite number m of square domains of finite breadths, so that every point of the domain is either within one of these square domains or on its boundary, and so that no point is within two of the square domains; also we may devise a rule by which, as the number m increases indefinitely, the breadths of all the square domains are diminished indefinitely. When this process is applied to a homogeneous part, H, of the numerical continuumCn, then, at any stage of the process, there will be some square domains of which all the points belong to H, and there will generally be others of which some, but not all, of the points belong to H. As the number m is increased indefinitely the sums of the extents of both these categories of square domains will tend to definite limits, which cannot be negative; when the second of these limits is zero the domain H is said to be “measurable,” and the first of these limits is its “extent”; it is independent of the rule adopted for constructing the square domains and contracting their breadths. The notion thus introduced may be adapted by suitable modifications to continua of lower dimensions inCn.