Thomae's function
{\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.
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)
It is a modification of the Dirichlet function, which is 1 at rational numbers and 0 elsewhere.
it suffices to check all irrational points in
According to the Archimedean property of the reals, there exists
to its i-th lower and upper bounds equals
then any choice of (sufficiently small)
Empirical probability distributions related to Thomae's function appear in DNA sequencing.
[7] 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 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.
In the case of uniform distributions on the set
, which is very similar to Thomae's function.
[7] For integers, the exponent of the highest power of 2 dividing
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 A001511 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.
If such a function existed, then the irrationals would be an Fσ set.
The irrationals would then be the countable union of closed sets
would be nowhere dense, and the irrationals would be a meager set.
It would follow that the real numbers, being the union of the irrationals and the rationals (which, as a countable set, 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 Fσ subset of the real numbers can be the set of discontinuities of a function.
is a countable union of closed sets
Then a similar argument as for Thomae's function shows that
