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

160 INTRODUCTION TO MATHEMATICS
of the function for arguments, other than a,
lying within that interval posessses the characteristic. The valuef(a) of the function for
the argument a may or may not possess the
characteristic. Nothing is decided on this
point by statements about the neighbourhood
of a.
For example, suppose we take the particular function x2. Now in the neighbourhood of 2, the values of x2 are less than 5.
For we can find an interval, e.g. from 1 to
2-1, which (i) contains 2 not as an end-point,
and (ii) is such that, for values of x lying
within it, x2 is less than 5.
Now, combining the preceding ideas we
know what is meant by saying that in the
neighbourhood of a the function f(x) approximates to c within the standard k. It means
that some interval can be found which (i) includes a not as an end-point, and (ii) is such
that all values of f(x), where x lies in the interval and is not a, differ from c by less than k.
For example, in the neighbourhood of 2, the
function / approximates to 1-41425 within
the standard.0001. This is true because the
square root of 1-99996164 is 1-4142 and the
square root of 2-00024449 is 1-4143; hence
for values of z lying in the interval
1-99996164 to 2-00024449, which contains 2
not as an end-point, the values of the function./x all lie between 1-4142 and 1-4143, and