Thomae's function

Thomae's function is a real-valued function of a real variable that can be defined as:^[1]: 531 $${\displaystyle f(x)={\begin{cases}{\frac {1}{q}}&{\text{if }}x={\tfrac {p}{q}}\quad (x{\text{ is rational), with }}p\in \mathbb {Z} {\text{ and }}q\in \mathbb {N} {\text{ coprime}}\\0&{\text{if }}x{\text{ is irrational.}}\end{cases}}}$$

It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused with the integer ruler function),^[2] the Riemann function, or the Stars over Babylon (John Horton Conway's name).^[3] Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration.^[4]

Since every rational number has a unique representation with coprime (also termed relatively prime) ${\displaystyle p\in \mathbb {Z} }$ and ${\displaystyle q\in \mathbb {N} }$, the function is well-defined. Note that ${\displaystyle q=+1}$ is the only number in ${\displaystyle \mathbb {N} }$ that is coprime to ${\displaystyle p=0.}$

It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere.