XI

0 , and thence through the series of increasing positive values. Accordingly, if a moving point, M , represents x on X O X ′ , M starts at the extreme left of the axis X O X ′ and successively moves through M 1 , M 2 , M 3 , M 4 , etc. The corresponding points on the function are P 1 , P 2 , P 3 , P 4 , etc. It is easy to see that there is a point of discontinuity at x = 0 , i.e. at the origin O . For the value of the function on the negative (left) side of the origin becomes endlessly great, but negative, and the function reappears on the positive (right) side as endlessly great but positive. Hence, however small we take the length M 2 M 3 , there is a finite jump between the values of the function at M 2 and M 3 . Indeed, this case has the peculiarity that the smaller we take the length between M 2 and M 3 , so long as they enclose the origin, the bigger is the jump in value of the function between them. This graph brings out, what is also apparent in [fig.]20 of this chapter, that for many functions the discontinuities only occur at isolated points, so that by restricting the values of the argument we obtain a continuous function for these remaining values. Thus it is evident

from [fig.]21 that in $y={\displaystyle \frac{1}{x}}$, if we keep to positive values only and exclude the origin, we obtain a continuous function. Similarly the same function, if we keep to negative values only, excluding the origin, is continuous. Again the function which is graphed in [fig.]20 is continuous between $B$ and $C_1$, and between $C_1$ and $C_2$, and between $C_2$ and $C_3$, and so on, always in each case excluding the end points. It is, however, easy to find functions such that their discontinuities occur at all points. For example, consider a function $f(x)$, such that when $x$ is any fractional number $f(x)=1$, and when $x$ is any incommensurable number $f(x)=2$. This function is discontinuous at all points.

Finally, we will look a little more closely at the definition of continuity given above. We have said that 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. This is exactly the sort of definition which satisfied our mathematical forefathers and no longer satisfies modern mathematicians. It is worth while to spend some time over it; for when we understand the modern objections to it, we shall have gone a long way towards the understanding of the spirit of modern mathematics. The

whole difference between the older and the newer mathematics lies in the fact that vague half-metaphorical terms like "gradually" are no longer tolerated in its exact statements. Modern mathematics will only admit statements and definitions and arguments which exclusively employ the few simple ideas about number and magnitude and variables on which the science is founded. Of two numbers one can be greater or less than the other; and one can be such and such a multiple of the other; but there is no relation of "graduality" between two numbers, and hence the term is inadmissible. Now this may seem at first sight to be great pedantry. To this charge there are two answers. In the first place, during the first half of the nineteenth century it was found by some great mathematicians, especially Abel in

Sweden, and Weierstrass in Germany, that

large parts of mathematics as enunciated in the old happy-go-lucky manner were simply wrong. Macaulay in his essay on Bacon

contrasts the certainty of mathematics with the uncertainty of philosophy; and by way of a rhetorical example he says, "There has been no reaction against Taylor's theorem."