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

FUNCTIONS 159
A set of numbers approximates to a number a within a standard k, when the numerical
difference between a and every number of the
set is less than k. Here k is the "standard
of approximation." Thus the set of numbers 3, 4, 6, 8, approximates to the number
5 within the standard 4. In this case the
standard 4 is not the smallest which could
have been chosen, the set also approximates
to 5 within any of the standards 3-1 or 3'01
or 3'001. Again, the numbers, 3-1, 3-141,
3-1415, 3-14159 approximate to 3-13102
within the standard -032, and also within the
smaller standard -03103.
These two ideas of an interval and of approximation to a number within a standard
are easy enough; their only difficulty is that
they look rather trivial. But when combined
with the next idea, that of the "neighbourhood" of a number, they form the foundation of modern mathematical reasoning.
What do we mean by saying that something
is true for a function f(x) in the neighbourhood of the value a of argument x? It is
this fundamental notion which we have now
got to make precise.
The values of a function f(x) are said to
possess a characteristic in the "neighbourhood of a" when some interval can be found,
which (i) contains the number a not as an
end-point, and (ii) is such that every value