: Mathematics: Thomae's function

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Thomae's function

Thomae's function is a real-valued function of a real variable that can be defined as:
${\\displaystyle f(x)={\egin{cases}{\ rac {1}{q}}&{\ ext{if }}x={\ frac {p}{q}}\\quad (x{\ ext{ is rational), with }}p\\in \\mathbb {Z} {\ ext{ and }}q\\in \\mathbb {N} {\ ext{ coprime}}\\\\0&{\ ext{if }}x{\ ext{ is irrational.}}\\end{cases}}}$
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, the Riemann function, or the Stars over Babylon (John Horton Conway'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.

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Thomae%27s_function)

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