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

FUNCTIONS 161
they therefore all differ from 1'41425 by less
than '0001. In this case we can, if we like,
fix a smaller standard of approximation,
namely *000051 or *0000501. Again, to take
another example, in the neighbourhood of 2
the function x2 approximates to 4 within the
standard -5. For (1-9)2=3 61 and (2.1)2=
4-41, and thus the required interval 1'9 to
2'1, containing 2 not as an end-point, has
been found. This example brings out the
fact that statements about a function f(x) in
the neighbourhood of a number a are distinct
from statements about the value off(x) when
x = a. The production of an interval, throughout which the statement is true, is required.
Thus the mere fact that 22 =4 does not by
itself justify us in saying that in the neighbourhood of 2 the function x2 is equal to 4.
This statement would be untrue, because no
interval can be produced with the required
property. Also, the fact that 22=4 does not
by itself justify us in saying that in the
neighbourhood of 2 the function x2 approximates to 4 within the standard '5; although
as a matter of fact, the statement has just
been proved to be true.
If we understand the preceding ideas, we
understand the foundations of modern
mathematics. We shall recur to analogous
ideas in the chapter on Series, and again
in the chapter on the Differential Calculus.