Microcontinuity

<h2 id="history">History</h2>
<p>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, <a href="/facts/Cauchy/aoTjgj6j">Cauchy</a>'s textbook <a href="/facts/Cours_d%27Analyse/IUUvmV23">Cours d'Analyse</a> defined continuity in 1821 using <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimals</a> as above.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="continuity-and-uniform-continuity">Continuity and uniform continuity</h2>
<p>The property of microcontinuity is typically applied to the natural extension <i>f*</i> of a real function <i>f</i>.  Thus, <i>f</i> defined on a real interval <i>I</i> is <a href="/facts/Continuous_function/sKbl02pB">continuous</a> if and only if <i>f*</i>  is microcontinuous at every point of <i>I</i>.  Meanwhile, <i>f</i> is <a href="/facts/Uniformly_continuous/hjvw7tII">uniformly continuous</a> on <i>I</i> if and only if <i>f*</i> is microcontinuous at every point (standard and nonstandard) of the natural extension <i>I*</i> of its domain <i>I</i> (see Davis, 1977, p. 96).
</p>
<h2 id="example-1">Example 1</h2>
<p>The real function 
  
    
      
        f
        (
        x
        )
        =
        
          
            
              1
              x
            
          
        
      
    
    {\displaystyle f(x)={\tfrac {1}{x}}}
  
 on the open interval (0,1) is not uniformly continuous because the natural extension <i>f*</i> of <i>f</i> fails to be microcontinuous at an <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> 
  
    
      
        a
        >
        0
      
    
    {\displaystyle a>0}
  
.  Indeed, for such an <i>a</i>, the values <i>a</i> and <i>2a</i> are infinitely close, but the values of <i>f*</i>, namely 
  
    
      
        
          
            
              1
              a
            
          
        
      
    
    {\displaystyle {\tfrac {1}{a}}}
  
 and 
  
    
      
        
          
            
              1
              
                2
                a
              
            
          
        
      
    
    {\displaystyle {\tfrac {1}{2a}}}
  
 are not infinitely close.
</p>
<h2 id="example-2">Example 2</h2>
<p>The function 
  
    
      
        f
        (
        x
        )
        =
        
          x
          
            2
          
        
      
    
    {\displaystyle f(x)=x^{2}}
  
 on 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 is not uniformly continuous because <i>f*</i> fails to be microcontinuous at an infinite point 
  
    
      
        H
        ∈
        
          
            R
          
          
            ∗
          
        
      
    
    {\displaystyle H\in \mathbb {R} ^{*}}
  
.  Namely, setting 
  
    
      
        e
        =
        
          
            
              1
              H
            
          
        
      
    
    {\displaystyle e={\tfrac {1}{H}}}
  
 and <i>K</i> = <i>H</i> + <i>e</i>, one easily sees that <i>H</i> and <i>K</i> are infinitely close but <i>f</i>*(<i>H</i>) and <i>f</i>*(<i>K</i>) are not infinitely close.
</p>
<h2 id="uniform-convergence">Uniform convergence</h2>
<p><a href="/facts/Uniform_convergence/ecz9eNWn">Uniform convergence</a> similarly admits a simplified definition in a hyperreal setting.  Thus, a sequence 
  
    
      
        
          f
          
            n
          
        
      
    
    {\displaystyle f_{n}}
  
 converges to <i>f</i> uniformly if for all <i>x</i> in the domain of <i>f*</i> and all infinite <i>n</i>, 
  
    
      
        
          f
          
            n
          
          
            ∗
          
        
        (
        x
        )
      
    
    {\displaystyle f_{n}^{*}(x)}
  
 is infinitely close to 
  
    
      
        
          f
          
            ∗
          
        
        (
        x
        )
      
    
    {\displaystyle f^{*}(x)}
  
.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Standard_part_function/fTuL5IXj">Standard part function</a></li></ul>
<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/Martin_Davis_(mathematician)/nhkpJl5N">Martin Davis</a> (1977) Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney. xii+181 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-19897-8</li>
<li>Gordon, E. I.; Kusraev, A. G.; <a href="/facts/Semen_Samsonovich_Kutateladze/Bq99vQUf">Kutateladze</a>, 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.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Borovik, Alexandre; Katz, Mikhail G. (2011), "Who gave you the Cauchy--Weierstrass tale? The dual history of rigorous calculus", Foundations of Science, 17 (3): 245–276, arXiv:1108.2885, doi:10.1007/s10699-011-9235-x. <a href="/wiki/Alexandre_Borovik" target="_blank">/wiki/Alexandre_Borovik</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Microcontinuity open-in-new

Microcontinuity