Thomae's function

Thomae's function, named after Carl Johannes Thomae, has many names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function,[1] the Riemann function, or the Stars over Babylon (John Horton Conway's name).[2] This real-valued function of a real variable can be defined as:[3]

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

Empirical probability distributions related to Thomae's function appear in DNA sequencing.[5] The human genome is diploid, having two strands per chromosome. When sequenced, small pieces ("reads") are generated: for each spot on the genome, an integer number of reads overlap with it. Their ratio is a rational number, and typically distributed similarly to Thomae's function.

For a general construction on arbitrary metric space, see this article