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

162 INTRODUCTION TO MATHEMATICS
Meanwhile, we are now prepared to define
"continuous functions." A function f(x)
is "continuous " at a value a of its argument, when in the neighbourhood of a
its values approximate to f(a) (i.e. to its
value at a) within every standard of approximation.
This means that, whatever standard k be
chosen, in the neighbourhood of a f(x) approximates to f(a) within the standard k.
For example, x2 is continuous at the value 2
of its argument, x, because however k be
chosen we can always find an interval, which
(i) contains 2 not as an end-point, and (ii) is
such that the values of x2 for arguments lying
within it approximate to 4 (i.e. 22) within
the standard k. Thus, suppose we choose
the standard -1; now (1.999)2 = 3996001,
and (2 01)2 =4 0401, and both these numbers
differ from 4 by less than '1. Hence, within
the interval 1-999 to 2-01 the values of x2
approximate to 4 within the standard '1.
Similarly an interval can be produced for any
other standard which we like to try.
Take the example of the railway train. Its
velocity is continuous as it passes the signal
box, if whatever velocity you like to assign
(say one-millionth of a mile per hour) an interval of time can be found extending before
and after the instant of passing, such that at
all instants within it the train's velocity