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

FUNCTIONS 155
in y= -, if we keep to positive values only
x
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 Ci, and between Ci and C2, and
between C2 and Cs, 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