Thomae's function

Thomae's function is a <a href="/facts/Real_number/R02gw5Pb">real</a>-valued <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> of a real variable that can be defined as:: 531

f
        (
        x
        )
        =
        
          
            {
            
              
                
                  
                    
                      1
                      q
                    
                  
                
                
                  
                    if 
                  
                  x
                  =
                  
                    
                      
                        p
                        q
                      
                    
                  
                  
                  (
                  x
                  
                     is rational), with 
                  
                  p
                  ∈
                  
                    Z
                  
                  
                     and 
                  
                  q
                  ∈
                  
                    N
                  
                  
                     coprime
                  
                
              
              
                
                  0
                
                
                  
                    if 
                  
                  x
                  
                     is irrational.
                  
                
              
            
            
          
        
      
    
    {\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac {p}{q}}\quad (x{\text{ is rational), with }}p\in \mathbb {Z} {\text{ and }}q\in \mathbb {N} {\text{ coprime}}\\0&{\text{if }}x{\text{ is irrational.}}\end{cases}}}

It is named after <a href="/facts/Carl_Johannes_Thomae/VbKzoJ14">Carl Johannes Thomae</a>, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified <a href="/facts/Dirichlet_function/d9o708HR">Dirichlet function</a>, the ruler function (not to be confused with the integer <a href="/facts/Ruler_function/SjCt2ahN">ruler function</a>), the Riemann function, or the Stars over Babylon (<a href="/facts/John_Horton_Conway/g3PA1zoK">John Horton Conway</a>'s name). Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration.
Since every <a href="/facts/Rational_number/VJQmyY05">rational number</a> has a unique representation with <a href="/facts/Coprime_integers/bnCxCXxs">coprime</a> (also termed relatively prime) 
 
 
 
 p
 ∈
 
 Z
 
 
 
 {\displaystyle p\in \mathbb {Z} }
 
 and 
 
 
 
 q
 ∈
 
 N
 
 
 
 {\displaystyle q\in \mathbb {N} }
 
, the function is <a href="/facts/Well-defined/GAc0XbWz">well-defined</a>. Note that 
 
 
 
 q
 =
 +
 1
 
 
 {\displaystyle q=+1}
 
 is the only number in 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbb {N} }
 
 that is coprime to 
 
 
 
 p
 =
 0.
 
 
 {\displaystyle p=0.}

It is a modification of the <a href="/facts/Dirichlet_function/d9o708HR">Dirichlet function</a>, which is 1 at rational numbers and 0 elsewhere.

Thomae's function open-in-new

Thomae's function