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

158 INTRODUCTION TO MATHEMATICS
pounds is a large yearly income. Again, onequarter is a large fraction of one's income to
give away in charity, but is a small fraction
of it to retain for private use. Examples can
be accumulated indefinitely to show that
great or small in any absolute sense have
no abstract application to numbers. We can
say of two numbers that one is greater or
smaller than another, but not without specification of particular circumstances that any
one number is great or small. Our task
therefore is to define continuity without any
mention of a "small" or "gradual" change
in value of the function.
In order to do this we will give names to
some ideas, which will also be useful when
we come to consider limits and the differential calculus.
An "interval" of values of the argument
x of a function f(x) is all the values lying
between some two values of the argument.
For example, the interval between x = 1 and
x =2 consists of all the values which x can
take lying between 1 and 2, i.e. it consists of
all the real numbers between 1 and 2. But
the bounding numbers of an interval need
not be integers. An interval of values of the
argument contains a number a, when a is a
member of the interval. For example, the
interval between 1 and 2 contains 1,,;, and
so on.