Chapter I: Functions of Real Variables (1)

1. _Historical._--The word function, defined in the above sense, was introduced by Leibnitz in a short note of date 1694 concerning the construction of what we now call an "envelope" (_Leibnizens mathematische Schriften_, edited by C.I. Gerhardt, Bd. v. p. 306), and was there used to denote a variable length related in a defined way to a variable point of a curve. In 1698 James Bernoulli used the word in a special sense in connexion with some isoperimetric problems (Joh. Bernoulli, _Opera_, t. i. p. 255). He said that when it is a question of selecting from an infinite set of like curves that one which best fulfils some function, then of two curves whose intersection determines the thing sought one is always the "line of the function" (_Linea functionis_). In 1718 John Bernoulli (_Opera_, t. ii. p. 241) defined a "function of a variable magnitude" as a quantity made up in any way of this variable magnitude and constants; and in 1730 (Opera, t. iii. p. 174) he noted a distinction between "algebraic" and "transcendental" functions. By the latter he meant integrals of algebraic functions. The notation [f](x) for a function of a variable x was introduced by Leonhard Euler in 1734 (_Comm. Acad. Petropol._ t. vii. p. 186), in connexion with the theorem of the interchange of the order of differentiations. The notion of functionality or functional relation of two magnitudes was thus of geometrical origin; but a function soon came to be regarded as an analytical expression, not necessarily an algebraic expression, containing the variable or variables. Thus we may have rational integral algebraic functions such as _ax_^2 + _bx_ + c, or rational algebraic functions which are not integral, such as

a1x^n + a2x^(n - 1) + ... + a_n
-------------------------------,
b1x^m + b2x^(m - 1) + ... + b_m

or irrational algebraic functions, such as [root]x, or, more generally the algebraic functions that are determined implicitly by an algebraic equation, as, for instance,

[f]_n(x, y) + [f]_(n - 1) (x, y) + ... + [f]0 = 0

where [f]_n(x, y), ... mean homogeneous expressions in x and y having constant coefficients, and having the degrees indicated by the suffixes, and [f]0 is a constant. Or again we may have trigonometrical functions, such as sin x and tan x, or inverse trigonometrical functions, such as sin^(-1)x, or exponential functions, such as e^x and a^x, or logarithmic functions, such as log x and log (1 + x). We may have these functional symbols combined in various ways, and thus there arises a great number of functions. Further we may have functions of more than one variable, as, for instance, the expression xy/(x^2 + y^2), in which both x and y are regarded as variable. Such functions were introduced into analysis somewhat unsystematically as the need for them arose, and the later developments of analysis led to the introduction of other classes of functions.

2. _Graphic Representation._--In the case of a function of one variable x, any value of x and the corresponding value y of the function can be the co-ordinates of a point in a plane. To any value of x there corresponds a point N on the axis of x, in accordance with the rule that x is the abscissa of N. The corresponding value of y determines a point P in accordance with the rule that x is the abscissa and y the ordinate of P. The ordinate y gives the value of the function which corresponds to that value of the variable x which is specified by N; and it may be described as "the value of the function at N." Since there is a one-to-one correspondence of the points N and the numbers x, we may also describe the ordinate as "the value of the function at x." In simple cases the aggregate of the points P which are determined by any particular function (of one variable) is a curve, called the "graph of the function" (see S 14). In like manner a function of two variables defines a surface.

3. _The Variable._--Graphic methods of representation, such as those just described, enabled mathematicians to deal with irrational values of functions and variables at the time when there was no theory of irrational numbers other than Euclid's theory of incommensurables. In that theory an irrational number was the ratio of two incommensurable geometric magnitudes. In the modern theory of number irrational numbers are defined in a purely arithmetical manner, independent of the measurement of any quantities or magnitudes, whether geometric or of any other kind. The definition is effected by means of the system of _ordinal_ numbers (see NUMBER). When this formal system is established, the theory of measurement may be founded upon it; and, in particular, the co-ordinates of a point are defined as numbers (not lengths), which are assigned in accordance with a rule. This rule involves the measurement of lengths. The theory of functions can be developed without any reference to graphs, or co-ordinates or lengths. The process by which analysis has been freed from any consideration of measurable quantities has been called the "arithmetization of analysis." In the theory so developed, the variable upon which a function depends is always to be regarded as a number, and the corresponding value of the function is also a number. Any reference to points or co-ordinates is to be regarded as a picturesque mode of expression, pointing to a possible application of the theory to geometry. The development of "arithmetized analysis" in the 19th century is associated with the name of Karl Weierstrass.

All possible values of a variable are numbers. In what follows we shall confine our attention to the case where the numbers are real. When complex numbers are introduced, instead of real ones, the theory of functions receives a wide extension, which is accompanied by appropriate limitations (see below, II. Functions of Complex Variables). The set of all real numbers forms a _continuum_. In fact the notion of a one-dimensional continuum first becomes precise in virtue of the establishment of the system of real numbers.

4. _Domain of a Variable.--Theory of Aggregates._--The notion of a "variable" is that of a number to which we may assign at pleasure any one of the values that belong to some chosen set, or _aggregate_, of numbers; and this set, or aggregate, is called the "domain of the variable." This domain may be an "interval," that is to say it may consist of two terminal numbers, all the numbers between them and no others. When this is the case the number is said to be "continuously variable." When the domain consists of all real numbers, the variable is said to be "unrestricted." A domain which consists of all the real numbers which exceed some fixed number may be described as an "interval unlimited towards the right"; similarly we may have an interval "unlimited towards the left."

In more complicated cases we must have some rule or process for
assigning the aggregate of numbers which constitute the domain of a
variable. The methods of definition of particular types of aggregates,
and the theorems relating to them, form a branch of analysis called
the "theory of aggregates" (_Mengenlehre, Theorie des ensembles,
Theory of sets of points_). The notion of an "aggregate" in general
underlies the system of ordinal numbers. An aggregate is said to be
"infinite" when it is possible to effect a one-to-one correspondence
of all its elements to some of its elements. For example, we may make
all the integers correspond to the even integers, by making 1
correspond to 2, 2 to 4, and generally n to 2n. The aggregate of
positive integers is an infinite aggregate. The aggregates of all
rational numbers and of all real numbers and of points on a line are
other examples of infinite aggregates. An aggregate whose elements are
real numbers is said to "extend to infinite values" if, after any
number N, however great, is specified, it is possible to find in the
aggregate numbers which exceed N in absolute value. Such an aggregate
is always infinite. The "neighbourhood of a number (or point) a for a
positive number h" is the aggregate of all numbers (or points) x for
which the absolute value of x - a denoted by |x - a|, does not exceed
h.

5. _General Notion of Functionality._--A function of one variable was for a long time commonly regarded as the ordinate of a curve; and the two notions (1) that which is determined by a curve supposed drawn, and (2) that which is determined by an analytical expression supposed written down, were not for a long time clearly distinguished. It was for this reason that Fourier's discovery that a single analytical expression is capable of representing (in different parts of an interval) what would in his time have been called different functions so profoundly struck mathematicians (S 23). The analysts who, in the middle of the 19th century, occupied themselves with the theory of the convergence of Fourier's series were led to impose a restriction on the character of a function in order that it should admit of such representation, and thus the door was opened for the introduction of the general notion of functional dependence. This notion may be expressed as follows: We have a variable number, y, and another variable number, x, a domain of the variable x, and a rule for assigning one or more definite values to y when x is any point in the domain; then y is said to be a "function" of the variable x, and x is called the "argument" of the function. According to this notion a function is, as it were, an indefinitely extended table, like a table of logarithms; to each point in the domain of the argument there correspond values for the function, but it remains arbitrary what values the function is to have at any such point.

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 values assigned
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 [f](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 [epsilon], however small, has been specified, it is possible to find a positive number h, so that |L - [f](x)| < [epsilon] for all points x of the domain (other than a) for which |x-a| < h, then L is the "limit of [f](x) at the point a." The condition for the existence of L is that, after the positive number [epsilon] has been specified, it must be possible to find a positive number h, so that |[f](x') - [f](x)| < [epsilon] 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 [f](x) at the point a as defined above. The limit of [f](x) at the point a is denoted by Lt_(x = a)[f](x), or by lim_(x = a)[f](x).

If [f](x) is a function of one variable x in a domain which extends to
infinite values, and if, after [epsilon] has been specified, it is
possible to find a number N, so that |[f](x') - [f](x)| <[epsilon] 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 |[f](x) - L| <
[epsilon] for all such values of x. In this case [f](x) has a limit L
at x = [oo]. In like manner [f](x) may have a limit at x = -[oo]. 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, ... u_n, ..., and this sequence can have a limit
at n = [oo].

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 = [oo]" 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 by _infinite series_.
When a sequence of numbers u1, u2, ... u_n, ... is given, we may form
a new sequence s1, s2, ... s_n, ... from it by the rules s1 = u1, s2 =
u1 + u2, ... s_n = u1 + u2 + ... + u_n or by the equivalent rules s1 =
u, s_n - s_(n - 1) = u_n(n = 2, 3, ...). If the new sequence has a
limit at n = [oo], this limit is called the "sum of the infinite
series" u1 + u2 + ..., and the series is said to be "convergent" (see
SERIES).

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/[pi], 1/2[pi], ... 1/n[pi], ... or to the aggregate
1/2[pi], 2/5[pi], ... n/(n^2 + 1)[pi], ..., 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 [f](a
- 0) and [f](a + 0).

The limit L of [f](x) at a stands in no necessary relation to the
value of [f](x) at a. If the point a is in the domain of the argument,
the value of [f](x) at a is assigned by the rule of calculation, and
may be different from L. In case [f](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 [f](x) at a. In the case where [f](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 [f](a) the value L; the function may then
be said to be "extrinsically defined." The so-called "indeterminate
forms" (see INFINITESIMAL 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 ([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.

We have also two theorems concerning the integral of the product of
two integrable functions [f](x) and [phi](x); these are known as "the
first and second theorems of the mean." The first theorem of the mean
is that, if [phi](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 [f](x) in the
interval, which has the property expressed by the equation
_ _
/ b / b
M | [phi](x)dx = | [f](x)[phi](x)dx.
_/a _/a

The second theorem of the mean is that, if [f](x) is monotonous
throughout the interval, there is a number [xi] between a and b which
has the property expressed by the equation
_ _ _
/ b /[xi] / b
| [f](x)[phi](x)dx = [f](a) | [phi](x)dx + [f](b) | [phi](x)dx.
_/a _/a _/[xi]

(_See_ FOURIER'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 [f](x) is not defined, we impose a restriction on the points x'_r of the definition: none of them is to be the point c. This comes to the same thing as defining [int] [a to b] [f](x)dx to be _ _ / c-[epsilon] / b Lt | [f](x)dx + Lt | [f](x)dx, (1) _/a _/c+[epsilon]' [epsilon]=0 [epsilon]'=0

where, to fix ideas, b is taken > a, and [epsilon] and [epsilon]' are positive. The same definition applies to the case where [f](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
_ _
/ [oo] / h
| [f](x)dx = Lt | [f](x)dx,
_/a _/a
h=[oo]

provided this limit exists. Similar definitions apply to
_ _
/-[oo] / [oo]
| [f](x)dx and to | [f](x)dx.
_/a _/-[oo]

All such definite integrals as the above are said to be "improper." For
example, [int] {0 to [oo]} (sin x / x)dx is improper in two ways. It
means
_
/ h sin x
Lt Lt | ----- dx,
h=[oo] [epsilon]=0 _/[epsilon] x

in which the positive number [epsilon] is first diminished indefinitely, and the positive number h is afterwards increased indefinitely.

The "theorems of the mean" (S 15) require modification when the integrals are improper (see FOURIER'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 [f](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 to b][f](x)_dx_, which has no value, is called a "divergent integral, "and it may happen that there is a definite value for _ _ _ _ | / c-[epsilon] / b | Lt | | [f](x) dx + | [f](x) dx | |_ _/a _/c+[epsilon]' _|

provided that [epsilon] and [epsilon]' 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 [epsilon]' = [epsilon], 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 (C_n) is the aggregate that is arrived at by attributing simultaneous values to each of n variables x1, x2, ... x_n, these values being any real numbers. The elements of such an aggregate are called "points," and the numbers x1, x2 ... x_n the "co-ordinates" of a point. Denoting in general the points (x1, x2, ... x_n) and (x'1, x'2 ... x'_n) by x and x', the sum of the differences |x1 - x'1| + |x2 - x'2| + ... + |x_n - 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 [epsilon], 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 [epsilon]. A perfect connected aggregate is a _continuum_. This is G. Cantor's definition.

The definition of a continuum in C_n 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" H_n of C_n. 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 C_n, which are not
points of H_n; these form a complementary aggregate H'_n. (2) There
are points "within" H_n; this means that for any such point there is a
neighbourhood consisting exclusively of points of H_n. (3) The points
of H_n which do not lie "within" H_n 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 H_n and points of
H'_n: the aggregate of these points is called the "boundary" of H_n.
(4) When any two points a, b within H_n are taken, it is possible to
find a number [epsilon] and a corresponding number m, and to choose
points x', x", ... x^m, so that the neighbourhood of a for [epsilon]
contains x', and consists exclusively of points within H_n, 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 C_n 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 H_n of C_n 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 [epsilon] 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 [f] has a limit at a if for any positive
[epsilon], however small, there is a corresponding h which has the
property that |[f](x') - [f](x)| < [epsilon], 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 [epsilon], however small,
has been specified, a number N can be found which is such that
|[f](x') - [f](x)| < [epsilon], 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 (S 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 (S 7). The
definition of a continuous function (S 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 {4tan^(-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 INFINITESIMAL CALCULUS).