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.