In Chapter. II we discussed the notion of a function of a real variable \(x\), and illustrated the discussion by a large number of examples of such functions. And the reader will remember that there was one important particular with regard to which the functions which we took as illustrations differed very widely. Some were defined for *all* values of \(x\), some for *rational* values only, some for *integral* values only, and so on.

Consider, for example, the following functions: (i) \(x\), (ii) \(\sqrt{x}\), (iii) the denominator of \(x\), (iv) the square root of the product of the numerator and the denominator of \(x\), (v) the largest prime factor of \(x\), (vi) the product of \(\sqrt{x}\) and the largest prime factor of \(x\), (vii) the \(x\)th prime number, (viii) the height measured in inches of convict \(x\) in Dartmoor prison.

Then the aggregates of values of \(x\) for which these functions are defined or, as we may say, the *fields of definition* of the functions, consist of (i) *all* values of \(x\), (ii) *all positive* values of \(x\), (iii) *all rational* values of \(x\), (iv) *all positive rational* values of \(x\), (v) *all integral* values of \(x\), (vi), (vii) *all positive integral* values of \(x\), (viii) a certain number of positive integral values of \(x\), viz., \(1\), \(2\), …, \(N\), where \(N\) is the total number of convicts at Dartmoor at a given moment of time.^{1}

Now let us consider a function, such as (vii) above, which is defined for all positive integral values of \(x\) and no others. This function may be regarded from two slightly different points of view. We may consider it, as has so far been our custom, as a function of the real variable \(x\) defined for some only of the values of \(x\), viz. positive integral values, and say that for all other values of \(x\) the definition fails. Or we may leave values of \(x\) other than positive integral values entirely out of account, and regard our function as a function of the *positive integral variable \(n\)*, whose values are the positive integers \[1,\ 2,\ 3,\ 4,\ \dots.\] In this case we may write \[y = \phi(n)\] and regard \(y\) now as a function of \(n\) defined for all values of \(n\).

It is obvious that any function of \(x\) defined for all values of \(x\) gives rise to a function of \(n\) defined for all values of \(n\). Thus from the function \(y = x^{2}\) we deduce the function \(y = n^{2}\) by merely omitting from consideration all values of \(x\) other than positive integers, and the corresponding values of \(y\). On the other hand from any function of \(n\) we can deduce any number of functions of \(x\) by merely assigning values to \(y\), corresponding to values of \(x\) other than positive integral values, in any way we please.

- In the last case \(N\) depends on the time, and convict \(x\), where \(x\) has a definite value, is a different individual at different moments of time. Thus if we take different moments of time into consideration we have a simple example of a function \(y = F(x, t)\) of two variables, defined for a certain range of values of \(t\), viz. from the time of the establishment of Dartmoor prison to the time of its abandonment, and for a certain number of positive integral values of \(x\), this number varying with \(t\).↩︎

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