Baire function

<h2 id="classification-of-baire-functions">Classification of Baire functions</h2>
<p>Baire functions of class α, for any countable <a href="/facts/Ordinal_number/GelFLjJt">ordinal number</a> α, form a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> of <a href="/facts/Real_number/R02gw5Pb">real</a>-valued functions defined on a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>, as follows.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<ul><li>The Baire class 0 functions are the <a href="/facts/Continuous_function_(topology)/sKbl02pB">continuous functions</a>.</li>
<li>The Baire class 1 functions are those functions which are the <a href="/facts/Pointwise_convergence/TXGpeyIK">pointwise limit</a> of a <a href="/facts/Sequence/gV2S4uqp">sequence</a> of Baire class 0 functions.</li>
<li>In general, the Baire class α functions are all functions which are the pointwise limit of a sequence of functions of Baire class less than α.</li></ul>
<p>Some authors define the classes slightly differently, by removing all functions of class less than α from the functions of class α. This means that each Baire function has a well defined class, but the functions of given class no longer form a vector space.
</p><p><a href="/facts/Henri_Lebesgue/ysngVGXc">Henri Lebesgue</a> proved that (for functions on the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a>) each Baire class of a countable ordinal number contains functions not in any smaller class, and that there exist functions which are not in any Baire class.
</p>
<h2 id="baire-class-1">Baire class 1</h2>
<p>Examples:
</p>
<ul><li>The <a href="/facts/Derivative/7Ito70Dh">derivative</a> of any <a href="/facts/Differentiable_function/jQqTgLk1">differentiable function</a> is of class 1. An example of a differentiable function whose derivative is not continuous (at <i>x</i> = 0) is the function equal to 
  
    
      
        
          x
          
            2
          
        
        sin
        ⁡
        (
        1
        
          /
        
        x
        )
      
    
    {\displaystyle x^{2}\sin(1/x)}
  
 when <i>x</i> ≠ 0, and 0 when <i>x</i> = 0. An infinite sum of similar functions (scaled and displaced by <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>) can even give a differentiable function whose derivative is discontinuous on a dense set. However, it necessarily has points of continuity, which follows easily from The Baire Characterisation Theorem (below; take <i>K</i> = <i>X</i> = R).</li>
<li>The characteristic function of the set of <a href="/facts/Integer/PUwwrawx">integers</a>, which equals 1 if <i>x</i> is an integer and 0 otherwise. (An infinite number of large discontinuities.)</li>
<li><a href="/facts/Thomae%2527s_function/SjCt2ahN">Thomae's function</a>, which is 0 for <a href="/facts/Irrational_number/Psv19TET">irrational</a> <i>x</i> and 1/<i>q</i> for a rational number <i>p</i>/<i>q</i> (in reduced form). (A dense set of discontinuities, namely the set of rational numbers.)</li>
<li>The characteristic function of the <a href="/facts/Cantor_set/6NUcMJvN">Cantor set</a>, which equals 1 if <i>x</i> is in the Cantor set and 0 otherwise. This function is 0 for an uncountable set of <i>x</i> values, and 1 for an uncountable set. It is discontinuous wherever it equals 1 and continuous wherever it equals 0. It is approximated by the continuous functions 
  
    
      
        
          g
          
            n
          
        
        (
        x
        )
        =
        max
        (
        0
        ,
        
          1
          −
          n
          d
          (
          x
          ,
          C
          )
        
        )
      
    
    {\displaystyle g_{n}(x)=\max(0,{1-nd(x,C)})}
  
, where 
  
    
      
        d
        (
        x
        ,
        C
        )
      
    
    {\displaystyle d(x,C)}
  
 is the distance of x from the nearest point in the Cantor set.</li></ul>
<p>The Baire Characterisation Theorem states that a real valued function <i>f</i> defined on a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> <i>X</i> is a Baire-1 function if and only if for every <a href="/facts/Empty_set/ffEk3eA6">non-empty</a> <a href="/facts/Metric_space/gnBBRXtc">closed</a> subset <i>K</i> of <i>X</i>, the <a href="/facts/Restriction_(mathematics)/JV59DhKy">restriction</a> of <i>f</i> to <i>K</i> has a point of continuity relative to the <a href="/facts/Topological_space/v2ut7CbK">topology</a> of <i>K</i>.
</p><p>By another theorem of Baire, for every Baire-1 function the points of continuity are a <a href="/facts/Comeager/AxTk8frw">comeager</a> <a href="/facts/G%25CE%25B4_set/Erg1BCFk"><i>G</i>δ set</a> (Kechris 1995, Theorem (24.14)).
</p>
<h2 id="baire-class-2">Baire class 2</h2>
<p>An example of a Baire class 2 function on the interval [0,1] that is not of class 1 is the characteristic function of the rational numbers, 
  
    
      
        
          χ
          
            
              Q
            
          
        
      
    
    {\displaystyle \chi _{\mathbb {Q} }}
  
, also known as the <a href="/facts/Dirichlet_function/d9o708HR">Dirichlet function</a> which is <a href="/facts/Nowhere_continuous_function/XbxlDzx1">discontinuous everywhere</a>.
</p>
Proof
<p>We present two proofs.
</p>
<ol><li>This can be seen by noting that for any finite collection of rationals, the characteristic function for this set is Baire 1: namely the function 
  
    
      
        
          g
          
            n
          
        
        (
        x
        )
        =
        max
        (
        0
        ,
        
          1
          −
          n
          d
          (
          x
          ,
          K
          )
        
        )
      
    
    {\displaystyle g_{n}(x)=\max(0,{1-nd(x,K)})}
  
 converges identically to the characteristic function of 
  
    
      
        K
      
    
    {\displaystyle K}
  
, where 
  
    
      
        K
      
    
    {\displaystyle K}
  
 is the finite collection of rationals. Since the rationals are countable, we can look at the pointwise limit of these things over 
  
    
      
        
          K
          
            n
          
        
        =
        {
        
          r
          
            1
          
        
        ,
        
          r
          
            2
          
        
        ,
        …
        ,
        
          r
          
            n
          
        
        }
      
    
    {\displaystyle K_{n}=\{r_{1},r_{2},\dots ,r_{n}\}}
  
, where 
  
    
      
        
          r
          
            n
          
        
      
    
    {\displaystyle r_{n}}
  
 is an enumeration of the rationals. It is not Baire-1 by the theorem mentioned above: the set of discontinuities is the entire interval (certainly, the set of points of continuity is not comeager).</li>
<li>The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:</li></ol>

∀
        x
        ∈
        
          R
        
        ,
        
        
          χ
          
            
              Q
            
          
        
        (
        x
        )
        =
        
          lim
          
            k
            →
            ∞
          
        
        
          (
          
            
              lim
              
                j
                →
                ∞
              
            
            
              
                (
                
                  cos
                  ⁡
                  (
                  k
                  !
                  π
                  x
                  )
                
                )
              
              
                2
                j
              
            
          
          )
        
      
    
    {\displaystyle \forall x\in \mathbb {R} ,\quad \chi _{\mathbb {Q} }(x)=\lim _{k\to \infty }\left(\lim _{j\to \infty }\left(\cos(k!\pi x)\right)^{2j}\right)}

for integer <i>j</i> and <i>k</i>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Baire_set/LuF74Wgr">Baire set</a></li>
<li><a href="/facts/Nowhere_continuous_function/XbxlDzx1">Nowhere continuous function</a></li></ul>

<h3>Inline</h3>

<h3>General</h3>
<ul><li>Baire, René-Louis (1899). <a href="https://archive.org/details/surlesfonctions00bairgoog"><i>Sur les fonctions de variables réelles</i></a> (Ph.D.) (in French). École Normale Supérieure.</li>
<li>Baire, René-Louis (1905), <i>Leçons sur les fonctions discontinues, professées au collège de France</i> (in French), Gauthier-Villars</li>
<li><a href="/facts/Alexander_S._Kechris/XUP7SToo">Kechris, Alexander S.</a> (1995), <i>Classical Descriptive Set Theory</i>, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4612-8692-9</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="https://encyclopediaofmath.org/wiki/Baire_classes">Springer Encyclopaedia of Mathematics article on Baire classes</a></li></ul>

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

<ol>
<li id="fn:1"><p>Jech, Thomas (November 1981). "The Brave New World of Determinacy". Bulletin of the American Mathematical Society. 5 (3): 339–349. doi:10.1090/S0273-0979-1981-14952-1. <a href="https://projecteuclid.org/journalArticle/Download?urlid=bams%2F1183548432" target="_blank">https://projecteuclid.org/journalArticle/Download?urlid=bams%2F1183548432</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Baire function open-in-new

Baire function