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Thomae's function
Function that is discontinuous at rationals and continuous at irrationals / From Wikipedia, the free encyclopedia
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Thomae's function is a real-valued function of a real variable that can be defined as:[1]: 531
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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,[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) and
, the function is well-defined. Note that
is the only number in
that is coprime to
It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere.