XI

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 $\mathrm{.0001}$. In this case we can, if we like, fix a smaller standard of approximation, namely $\mathrm{.000051}$ or $\mathrm{.0000501}$. Again, to take another example, in the neighbourhood of $2$ the function $x^2$ approximates to $4$ within the standard $\mathrm{.5}$. For $(1.9{)}^2=3.61$ and $(2.1{)}^2=4.41$, and thus the required interval $1.9$ to $2.1$, containing $2$ not as an end-point, has been found. This example brings out the fact that statements about a function $f(x)$ in the neighbourhood of a number $a$ are distinct from statements about the value of $f(x)$ when $x=a$. The production of an interval, throughout which the statement is true, is required. Thus the mere fact that $2^2=4$ does not by itself justify us in saying that in the neighbourhood of $2$ the function $x^2$ is equal to $4$. This statement would be untrue, because no interval can be produced with the required property. Also, the fact that $2^2=4$ does not by itself justify us in saying that in the neighbourhood of $2$ the function $x^2$ approximates to $4$ within the standard $\mathrm{.5}$; although as a matter of fact, the statement has just been proved to be true.

If we understand the preceding ideas, we understand the foundations of modern mathematics. We shall recur to analogous ideas in the chapter on Series, and again in the chapter on the Differential Calculus.

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, $x^2$ 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 $x^2$ for arguments lying within it approximate to $4$ (i.e. $2^2$) within the standard $k$. Thus, suppose we choose the standard $\mathrm{.1}$; now $(1.999{)}^2=3.996001$, and $(2.01{)}^2=4.0401$, and both these numbers differ from $4$ by less than $\mathrm{.1}$. Hence, within the interval $1.999$ to $2.01$ the values of $x^2$ approximate to $4$ within the standard $\mathrm{.1}$. Similarly an interval can be produced for any other standard which we like to try.