253
Views
4
CrossRef citations to date
0
Altmetric
Articles

Defining continuity of real functions of real variables

Pages 189-204 | Published online: 11 Apr 2016
 

Abstract

Continuity of a real function of a real variable has been defined in various ways over almost 200 years. Contrary to popular belief, the definitions are not all equivalent, because their consequences for four somewhat pathological functions reveal five essentially different cases. The four defensible ones imply just two cases for continuity on an interval if that is defined by using pointwise continuity at each point. Some authors had trouble: two different textbooks each gave two arguably inconsistent definitions, three more changed their definitions in their second editions, two more claimed continuity at a point for functions not defined there, and one gave a definition implying it for a function with no limit there.

Acknowledgements

I am grateful to Dr June Barrow-Green for helpful comments on this work and for Rice and Wilson Citation(2003), to the Cambridge University Library for access to rare publications, and to the Victoria University of Wellington for allowing me to use its facilities after retirement.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 Page numbers for definitions of continuity are given in the third section ‘Continuity of four functions’, either in citations or in square brackets in quotations. In a quotation, a row of dots not in square brackets is in the original text, but [...] indicates an omission.

Log in via your institution

Log in to Taylor & Francis Online

PDF download + Online access

  • 48 hours access to article PDF & online version
  • Article PDF can be downloaded
  • Article PDF can be printed
USD 61.00 Add to cart

Issue Purchase

  • 30 days online access to complete issue
  • Article PDFs can be downloaded
  • Article PDFs can be printed
USD 185.00 Add to cart

* Local tax will be added as applicable

Related Research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.