XI

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.

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

of the function for arguments, other than $a$, lying within that interval possesses the characteristic. The value $f(a)$ of the function for the argument $a$ may or may not possess the characteristic. Nothing is decided on this point by statements about the neighbourhood of $a$.

For example, suppose we take the particular function $x^2$. Now in the neighbourhood of $2$, the values of $x^2$ are less than $5$. For we can find an interval, e.g. from $1$ to $2.1$, which (i) contains $2$ not as an end-point, and (ii) is such that, for values of $x$ lying within it, $x^2$ is less than $5$.

Now, combining the preceding ideas we know what is meant by saying that in the neighbourhood of $a$ the function $f(x)$ approximates to $c$ within the standard $k$. It means that some interval can be found which (i) includes $a$ not as an end-point, and (ii) is such that all values of $f(x)$, where $x$ lies in the interval and is not $a$, differ from $c$ by less than $k$. For example, in the neighbourhood of $2$, the function $\sqrt{x}$ approximates to $1.41425$ within the standard $\mathrm{.0001}$. This is true because the square root of $1.99996164$ is $1.4142$ and the square root of $2.00024449$ is $1.4143$; hence for values of $x$ lying in the interval $1.99996164$ to $2.00024449$, which contains $2$ not as an end-point, the values of the function $\sqrt{x}$ all lie between $1.4142$ and $1.4143$, and

they therefore all differ from 1.41425 by less than .0001 . In this case we can, if we like, fix a smaller standard of approximation, namely .000051 or .0000501 . Again, to take another example, in the neighbourhood of 2