XI

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 $\frac{3}{2}$, $\frac{5}{3}$, $\frac{7}{4}$, and so on.

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 $\mathrm{.032}$, and also within the smaller standard $\mathrm{.03103}$.

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 the argument $x$? It is this fundamental notion which we have now got to make precise.