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CHAPTER IV - LIMITS OF FUNCTIONS OF A POSITIVE INTEGRAL VARIABLE

Published online by Cambridge University Press:  05 June 2012

G. H. Hardy
Foreword by
T. W. Körner
G. H. Hardy
Affiliation:
Trinity College, Cambridge
T. W. Körner
Affiliation:
University of Cambridge
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Summary

Functions of a positive integral variable. In Ch. 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) √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 √x and the largest prime factor of x, (vii) the xth 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.

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A Course of Pure Mathematics , pp. 110 - 171
Publisher: Cambridge University Press
Print publication year: 2008

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