World Library  
Flag as Inappropriate
Email this Article

Microcontinuity

Article Id: WHEBN0033952467
Reproduction Date:

Title: Microcontinuity  
Author: World Heritage Encyclopedia
Language: English
Subject: Non-standard analysis, (ε, δ)-definition of limit, Standard part function, Continuous function, Infinitesimals
Collection: Non-Standard Analysis
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Microcontinuity

In non-standard analysis, a discipline within classical mathematics, microcontinuity (or S-continuity) of an internal function f at a point a is defined as follows:

for all x infinitely close to a, the value f(x) is infinitely close to f(a).

Here x runs through the domain of f. In formulas, this can be expressed as follows:

if x\approx a then f(x)\approx f(a).

For a function f defined on \mathbb{R}, the definition can be expressed in terms of the halo as follows: f is microcontinuous at c\in\mathbb{R} if and only if f(hal(c))\subset hal(f(c)).

Contents

  • History 1
  • Continuity and uniform continuity 2
  • Example 1 3
  • Example 2 4
  • Uniform convergence 5
  • See also 6
  • Bibliography 7
  • References 8

History

The modern property of continuity of a function was first defined by Bolzano in 1817. However, Bolzano's work was not noticed by the larger mathematical community until its rediscovery in Heine in the 1860s. Meanwhile, Cauchy's textbook Cours d'Analyse defined continuity in 1821 using infinitesimals as above.[1]

Continuity and uniform continuity

The property of microcontinuity is typically applied to the natural extension f* of a real function f. Thus, f defined on a real interval I is continuous if and only if f* is microcontinuous at every point of I. Meanwhile, f is uniformly continuous on I if and only if f* is microcontinuous at every point (standard and non-standard) of the natural extension I* of its domain I (see Davis, 1977, p. 96).

Example 1

The real function f(x)=\frac{1}{x} on the open interval (0,1) is not uniformly continuous because the natural extension f* of f fails to be microcontinuous at an infinitesimal a>0. Indeed, for such an a, the values a and 2a are infinitely close, but the values of f*, namely \frac{1}{a} and \frac{1}{2a} are not infinitely close.

Example 2

The function f(x)=x^2 on \mathbb{R} is not uniformly continuous because f* fails to be microcontinuous at an infinite point H\in \mathbb{R}^*. Namely, setting e=\frac{1}{H} and K = H + e, one easily sees that H and K are infinitely close but f*(H) and f*(K) are not infinitely close.

Uniform convergence

Uniform convergence similarly admits a simplified definition in a hyperreal setting. Thus, a sequence f_n converges to f uniformly if for all x in the domain of f* and all infinite n, f_n^*(x) is infinitely close to f^*(x).

See also

Bibliography

  • Martin Davis (1977) Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney. xii+181 pp. ISBN 0-471-19897-8
  • Gordon, E. I.; Kusraev, A. G.; Kutateladze, S. S.: Infinitesimal analysis. Updated and revised translation of the 2001 Russian original. Translated by Kutateladze. Mathematics and its Applications, 544. Kluwer Academic Publishers, Dordrecht, 2002.

References

  1. ^  .
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.