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 provided this limit exists. Similar definitions apply to $\int_a^{-\infty}f(x)dx$, and to $\int_{-\infty}^{\infty}f(x)dx$. All such definite integrals as the above are said to be “improper.” For example, $$\int_0^{-\infty}\frac{\sin x}{x}dx$$ is improper in two ways. It means

in which the positive number 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 (see ).

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 $$\int_a^b f(x)dx$$, which has no value, is called a “divergent integral,” and it may happen that there is a definite value for

$Lt \bigg\{ \int_a^{c-\epsilon} f(x)dx + \int_{c+\epsilon '}^b fx(dx) \bigg\}$

provided that and ′ 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 ′ = , 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 xn the “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 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 , however small, a number m and points x′, x″, x(m) of the aggregate can be found so that all the differences denoted by aggregate is a continuum. This is G. Cantor’s definition.
 * x−a| <h. If an infinite aggregate of points does not extend
 * x′−a|,|x″−x′|, |b−x(m)| are less than . A perfect connected

The definition of a continuum in Cn leaves 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” Hn of 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 Hn which do not lie “within” Hn are 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 Hn and points of H′n: the aggregate of these points is called the “boundary” of Hn. (4) When any two points a, b within Hn are taken, it is possible to find a number and a corresponding number m, and to choose points x′, x″, x(m), so that the neighbourhood of a for 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 Cn when 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 Hn of Cn with 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 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, however small, there is a corresponding h which has the property that |ƒ(x′)−ƒ(x)| < , 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, however small, has been specified, a number N can be found which is such that |ƒ(x′)−ƒ(x)| < , 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 x1 and x2 may have a limit at (x1 = 0, x2 = 0) if we first diminish x1 without limit, keeping x2 constant, and afterwards diminish x2 without 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 (see ).

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| < h, 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 the