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where $f_{n}(x,y)$, $\ldots \$ 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 §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 gaphs, or co-ordinates or lengths. The process by which analysis has been freed from any consideration of measurable quantities has ben 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, Théorie 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 (§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 polnts 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