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

230 INTRODUCTION TO MATHEMATICS
A function f(x) is continuous at a value a
of its argument, when in the neighbourhood
of a its values approximate to its value at a
within every standard of approximation.
It is at once evident that a function is continuous at a when (i) it possesses a limit at a,
and (ii) that limit is equal to its value at a.
Thus the illustrations of continuity which
have been given at the end of Chapter XI are
illustrations of the idea of a limit, namely,
they were all directed to proving that f(a)
was the limit of f(x) at a for the functions
considered and the value of a considered. It
is really more instructive to consider the
limit at a point where a function is not continuous. For example, consider the function
of which the graph is given in fig. 20 of Chapter XI. This function f(x) is defined to have
the value 1 for all values of the argument
except the integers 1, 2, 3, etc., and for these
integral values it has the value 0. Now let
us think of its limit when x =3. We notice
that in the definition of the limit the value
of the function at a (in this case, a =3) is excluded. But, excluding f(3), the values of
f(x), when x lies within any interval which
(i) contains 3 not as an end-point, and (ii)
does not extend so far as 2 and 4, are all
equal to 1; and hence these values approximate to 1 within every standard of approximation. Hence 1 is the limit of f(x) at the