Dirichlet function

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Dirichlet function is the <a href="/facts/Indicator_function/QEuh04NM">indicator function</a> 
  
    
      
        
          
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    {\displaystyle \mathbf {1} _{\mathbb {Q} }}
  
 of the set of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> 
  
    
      
        
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    {\displaystyle \mathbb {Q} }
  
, i.e. 
  
    
      
        
          
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    {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=1}
  
 if x is a rational number and 
  
    
      
        
          
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    {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)=0}
  
 if x is not a rational number (i.e. is an <a href="/facts/Irrational_number/Psv19TET">irrational number</a>).

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    {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)={\begin{cases}1&x\in \mathbb {Q} \\0&x\notin \mathbb {Q} \end{cases}}}

</p><p>It is named after the mathematician <a href="/facts/Peter_Gustav_Lejeune_Dirichlet/b5UIU7Uy">Peter Gustav Lejeune Dirichlet</a>. It is an example of a <a href="/facts/Pathological_(mathematics)/KmKHjJ1H">pathological function</a> which provides counterexamples to many situations.
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Dirichlet function open-in-new

Dirichlet function