Dirichlet function

<h2 id="topological-properties">Topological properties</h2>
<ul><li>The Dirichlet function is <a href="/facts/Nowhere_continuous_function/XbxlDzx1">nowhere continuous</a>.
Proof
<ul><li>If y is rational, then f(y) = 1. To show the function is not continuous at y, we need to find an ε such that no matter how small we choose δ, there will be points z within δ of y such that f(z) is not within ε of f(y) = 1.  In fact, 1⁄2 is such an ε. Because the <a href="/facts/Irrational_number/Psv19TET">irrational numbers</a> are <a href="/facts/Dense_set/nPT9Yxao">dense</a> in the reals, no matter what δ we choose we can always find an irrational z within δ of y, and f(z) = 0 is at least 1⁄2 away from 1.</li>
<li>If y is irrational, then f(y) = 0. Again, we can take ε = 1⁄2, and this time, because the rational numbers are dense in the reals, we can pick z to be a rational number as close to y as is required. Again, f(z) = 1 is more than 1⁄2 away from f(y) = 0.</li></ul>

Its restrictions to the set of rational numbers and to the set of irrational numbers are <a href="/facts/Constant_function/BmPx6dn7">constants</a> and therefore continuous. The Dirichlet function is an archetypal example of the <a href="/facts/Blumberg_theorem/h015Form">Blumberg theorem</a>.</li><li>The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:

∀
        x
        ∈
        
          R
        
        ,
        
        
          
            1
          
          
            
              Q
            
          
        
        (
        x
        )
        =
        
          lim
          
            k
            →
            ∞
          
        
        
          (
          
            
              lim
              
                j
                →
                ∞
              
            
            
              
                (
                
                  cos
                  ⁡
                  (
                  k
                  !
                  π
                  x
                  )
                
                )
              
              
                2
                j
              
            
          
          )
        
      
    
    {\displaystyle \forall x\in \mathbb {R} ,\quad \mathbf {1} _{\mathbb {Q} }(x)=\lim _{k\to \infty }\left(\lim _{j\to \infty }\left(\cos(k!\pi x)\right)^{2j}\right)}

for integer j and k. This shows that the Dirichlet function is a <a href="/facts/Baire_function/P8sm52UD">Baire class</a> 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a <a href="/facts/Meagre_set/AxTk8frw">meagre set</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li></ul>
<h2 id="periodicity">Periodicity</h2>
<p>For any real number x and any positive rational number T, 
  
    
      
        
          
            1
          
          
            
              Q
            
          
        
        (
        x
        +
        T
        )
        =
        
          
            1
          
          
            
              Q
            
          
        
        (
        x
        )
      
    
    {\displaystyle \mathbf {1} _{\mathbb {Q} }(x+T)=\mathbf {1} _{\mathbb {Q} }(x)}
  
. The Dirichlet function is therefore an example of a real <a href="/facts/Periodic_function/Sm8lJQsF">periodic function</a> which is not <a href="/facts/Constant_function/BmPx6dn7">constant</a> but whose set of periods, the set of rational numbers, is a <a href="/facts/Dense_set/nPT9Yxao">dense subset</a> of 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
.
</p>
<h2 id="integration-properties">Integration properties</h2>
<ul><li>The Dirichlet function is not <a href="/facts/Riemann_integral/vWGrPhVh">Riemann-integrable</a> on any segment of 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 despite being bounded because the set of its discontinuity points is not <a href="/facts/Negligible_set/NIKJ6qCy">negligible</a> (for the <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a>).</li><li>The Dirichlet function has both an upper <a href="/facts/Darboux_integral/YgCMPJH9">Darboux integral</a> (namely, 
  
    
      
        b
        −
        a
      
    
    {\displaystyle b-a}
  
) and a lower Darboux integral (0) over any bounded interval 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
 — but they are not equal if 
  
    
      
        a
        <
        b
      
    
    {\displaystyle a<b}
  
, so the Dirichlet function is not Darboux-integrable (and therefore not Riemann-integrable) over any nondegenerate interval.</li><li>The Dirichlet function provides a counterexample showing that the <a href="/facts/Monotone_convergence_theorem/mYFH9yGY">monotone convergence theorem</a> is not true in the context of the Riemann integral.
Proof
<p>Using an <a href="/facts/Enumeration/EFmP9zL6">enumeration</a> of the rational numbers between 0 and 1, we define the function fn (for all nonnegative integer n) as the indicator function of the set of the first n terms of this sequence of rational numbers. The increasing sequence of functions fn (which are nonnegative, Riemann-integrable with a vanishing integral) pointwise converges to the Dirichlet function which is not Riemann-integrable.
</p>
</li><li>The Dirichlet function is <a href="/facts/Lebesgue_integral/f4AgvC6x">Lebesgue-integrable</a> on 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 and its integral over 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 is zero because it is zero except on the set of rational numbers which is negligible (for the Lebesgue measure).</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Thomae%27s_function/SjCt2ahN">Thomae's function</a>, a variation that is discontinuous only at the rational numbers</li></ul>

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

<ol>
<li id="fn:1"><p>"Dirichlet-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] <a href="https://www.encyclopediaofmath.org/index.php?title=Dirichlet-function" target="_blank">https://www.encyclopediaofmath.org/index.php?title=Dirichlet-function</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Dirichlet Function — from MathWorld <a href="http://mathworld.wolfram.com/DirichletFunction.html" target="_blank">http://mathworld.wolfram.com/DirichletFunction.html</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Lejeune Dirichlet, Peter Gustav (1829). "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données". Journal für die reine und angewandte Mathematik. 4: 157–169. <a href="https://eudml.org/doc/183134" target="_blank">https://eudml.org/doc/183134</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Dunham, William (2005). The Calculus Gallery. Princeton University Press. p. 197. ISBN 0-691-09565-5. <a href="0-691-09565-5" target="_blank">0-691-09565-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Dirichlet function open-in-new

Dirichlet function