Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-6bf8c574d5-b4m5d Total loading time: 0 Render date: 2025-02-16T05:42:23.745Z Has data issue: false hasContentIssue false

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
Get access

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×