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Point plot on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2
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, the Riemann function, or the Stars over Babylon (John Horton Conway's name). This real-valued function of a real variable can be defined as:
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.
The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero. Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval. The function's integral is equal to over any set because the function is equal to zero almost everywhere.
Related probability distributions
Empirical probability distributions related to Thomae's function appear in DNA sequencing. 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.
If pairs of positive integers are sampled from a distribution and used to generate ratios , this gives rise to a distribution on the rational numbers. If the integers are independent the distribution can be viewed as a convolution over the rational numbers, . Closed form solutions exist for power-law distributions with a cut-off. If (where is the polylogarithm function) then . In the case of uniform distributions on the set , which is very similar to Thomae's function. Both their graphs have fractal dimension 3/2.
The ruler function
For integers, the exponent of the highest power of 2 dividing gives 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, ... (sequence in the OEIS). If 1 is added, or if the 0s are removed, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 3, 1, 2, 1, ... (sequence in the OEIS). The values resemble tick-marks on a 1/16th graduated ruler, hence the name. These values correspond to the restriction of the Thomae function to the dyadic rationals: those rational numbers whose denominators are powers of 2.
A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible; the set of discontinuities of any function must be an F? set. If such a function existed, then the irrationals would be an F? set. The irrationals would then be the countable union of closed sets, but since the irrationals do not contain an interval, nor can any of the . Therefore, each of the would be nowhere dense, and the irrationals would be a meager set. It would follow that the real numbers, being a union of the irrationals and the rationals (which is evidently meager), would also be a meager set. This would contradict the Baire category theorem: because the reals form a complete metric space, they form a Baire space, which cannot be meager in itself.
A variant of Thomae's function can be used to show that any F? subset of the real numbers can be the set of discontinuities of a function. If is a countable union of closed sets , define
Then a similar argument as for Thomae's function shows that has A as its set of discontinuities.
^"...the so-called ruler function, a simple but provocative example that appeared in a work of Johannes Karl Thomae ... The graph suggests the vertical markings on a ruler--hence the name." (Dunham 2008, p. 149, chapter 10)
Beanland, Kevin; Roberts, James W.; Stevenson, Craig (2009), "Modifications of Thomae's Function and Differentiability", The American Mathematical Monthly, 116 (6): 531-535, doi:10.4169/193009709x470425, JSTOR40391145