With these explanations and cautions, we write $y = f (x)$ , to denote that $y$ is the value of some undetermined function of the argument $x$ ; where $f (x)$ may stand for anything such as $x + 1$ , $x^{2} - 2 x + 1$ , $\sin x$ , $\log x$ , or merely for $x$ itself. The essential point is that when $x$ is given, then $y$ is thereby definitely determined. It is important to be quite clear as to the generality of this idea. Thus in $y = f (x)$ , we may determine, if we choose, $f (x)$ to mean that when $x$ is an integer, $f (x)$ is zero, and when $x$ has any other value, $f (x)$ is $1$ . Accordingly, putting $y = f (x)$ , with this choice

for the meaning of $f$ , $y$ is either $0$ or $1$ according as the value of $x$ is integral or otherwise. Thus $f (1) = 0$ , $f (2) = 0$ , $f (\frac{2}{3}) = 1$ , $f (\sqrt{2}) = 1$ , and so on. This choice for the meaning of $f (x)$ gives a perfectly good function of the argument $x$ according to the general definition of a function.

A function, which after all is only a sort

of correlation between two variables, is represented like other correlations by a graph, that is in effect by the methods of coordinate geometry. For example, [fig.]2 in II. is the graph of the function $\frac{1}{v}$ where $v$ is the argument and $p$ the value of the function. In this case the graph is only drawn for positive values of $v$ , which are the only values possessing any meaning for the physical application considered in that chapter. Again in [fig.]14 of IX. the whole length of the line $A B$ , unlimited in both directions, is the graph of the function $x + 1$ , where $x$ is the argument and $y$ is the value of the function; and in the same figure the unlimited line $A_{1} B$ is the graph of the function $1 - x$ , and the line $L O L^{'}$ is the graph of the function $x$ , $x$ being the argument and $y$ the value of the function.

These functions, which are expressed by simple algebraic formulæ, are adapted for representation by graphs. But for some functions

this representation would be very misleading without a detailed explanation, or might even be impossible. Thus, consider the function mentioned above, which has the value $1$ for all values of its argument $x$ , except those which are integral, e.g. except for $x = 0$ , $x = 1$ , $x = 2$ , etc., when it has the value $0$ . Its appearance on a graph would be that of the straight line $A B A^{'}$ drawn parallel to the 20 axis $X O X^{'}$ at a distance from it of $1$ unit of length. But the points, $B$ , $C_{1}$ , $C_{2}$ , $C_{3}$ , $C_{4}$ , etc., corresponding to the values $0$ , $1$ , $2$ , $3$ , $4$ , etc., of the argument $x$ , are to be omitted, and instead of them the points $O$ , $B_{1}$ , $B_{2}$ , $B_{3}$ , $B_{4}$ , etc., on the axis $O X$ , are to be taken. It is easy to find functions for which the graphical representation is not only inconvenient but impossible. Functions which do not lend themselves to graphs are important in the

higher mathematics, but we need not concern ourselves further about them here.

The most important division between functions

is that between continuous and discontinuous functions. A function is continuous when its value only alters gradually for gradual alterations of the argument, and is discontinuous when it can alter its value by sudden jumps. Thus the two functions $x + 1$ and $1 - x$ , whose graphs are depicted as straight lines in [fig.]14 of IX., are continuous functions, and so is the function $\frac{1}{v}$ , depicted in II., if we only think of positive values of $v$ . But the function depicted in [fig.]20 of this chapter is discontinuous since at the values $x = 1$ , $x = 2$ , etc., of its argument, its value gives sudden jumps.