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::The [[Lebesgue integrability condition|Lebesgue criterion for integrability]] states that a bounded function is Riemann integrable if and only if the set of all discontinuities has [[Lebesgue measure|measure zero]].<ref>{{Harvnb|Spivak|1965|p=53|loc=Theorem 3-8}}</ref> Every [[countability|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 <math>0</math> over any set because the function is equal to zero ''[[almost everywhere]]''.
*If <math>G = \{ \, (x,f(x)) : x \in (0,1) \, \} \subset \mathbb{R}^2</math> is the graph of the restriction of <math>f</math> to <math>(0,1)</math>, then the [[Minkowski–Bouligand dimension|'''box-counting dimension''']] of <math>G</math> is <math>4/3</math>.<ref>{{cite journal |last1=Chen |first1=Haipeng |last2=Fraser |first2=Jonathan M. |last3=Yu |first3=Han |year=2022 |title=Dimensions of the popcorn graph |journal=Proceedings of the American Mathematical Society |volume=150 |number=11 |pages=4729–4742 |doi=10.1090/proc/15729 |arxiv=2007.08407}}</ref>
==Related probability distributions==
Empirical probability distributions related to Thomae's function appear in [[DNA sequencing]].<ref name="Trifonov">{{cite journal |last1=Trifonov |first1=Vladimir |last2=Pasqualucci |first2=Laura |last3=Dalla-Favera |first3=Riccardo |last4=Rabadan |first4=Raul |year=2011 |title=Fractal-like Distributions over the Rational Numbers in High-throughput Biological and Clinical Data |journal=Scientific Reports |volume=1 |number=191 |page=191 |doi=10.1038/srep00191 |pmid=22355706 |pmc=3240948|arxiv=1010.4328 |bibcode=2011NatSR...1E.191T }}</ref> 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 <math>m,n</math> are sampled from a distribution <math>f(n,m)</math> and used to generate ratios <math>q=n/(n+m)</math>, this gives rise to a distribution <math>g(q)</math> on the rational numbers. If the integers are independent the distribution can be viewed as a [[convolution]] over the rational numbers, <math display="inline">g(a/(a+b))=\sum_{t=1}^\infty f(ta)f(tb)</math>. Closed form solutions exist for [[power-law]] distributions with a cut-off. If <math>f(k)=k^{-\alpha} e^{-\beta k}/\mathrm{Li}_\alpha(e^{-\beta})</math> (where <math>\mathrm{Li}_\alpha</math> is the [[polylogarithm]] function) then <math>g(a/(a+b)) = (ab)^{-\alpha} \mathrm{Li}_{2\alpha}(e^{-(a+b)\beta})/\mathrm{Li}^2_{\alpha}(e^{-\beta})</math>. In the case of uniform distributions on the set <math>\{1,2,\ldots , L\}</math> <math>g(a/(a+b)) = (1/L^2) \lfloor L/\max(a,b) \rfloor</math>, which is very similar to Thomae's function
==The ruler function==
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