An introduction to mathematics, by A. N. Whitehead.

154 INTRODUCTION TO MATHEMATICS
now let x have any value, positive or negative.
The graph of the function is exhibited in fig.
21. Suppose x to change continuously from
a large negative value through a numerically
decreasing set of negative values up to 0, and
thence through the series of increasing positive values. Accordingly, if a moving point,
M, represents x on XOX', M starts at the
extreme left of the axis XOX' and successively moves through Mi, M2, M3, M4, etc.
The corresponding points on the function are
P1, P P3, P4, etc. It is easy to see that
there is a point of discontinuity at x = 0, i.e.
at the origin 0. 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 M2, M3, there is a
finite jump between the values of the function at M2 and M3. Indeed, this case has
the peculiarity that the smaller we take M2
M3, 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