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[[File:Thomae function (0,1).svg|200px|right|thumb|Point plot on the [[interval (mathematics)|interval]] (0,1). The topmost point in the middle shows ''f''(1/2) = 1/2]]
'''Thomae's function''' is a [[real number|real]]-valued [[function (mathematics)|function]] of a real variable that can be defined as:<ref name="Beanland">{{
<math display="block">f(x) =
\begin{cases}
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\end{cases}</math>
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''',<ref>{{Citation |last=Dunham |first=William |author-link=William Dunham (mathematician) |year=2008 |title=The Calculus Gallery: Masterpieces from Newton to Lebesgue |publisher=Princeton University Press |___location=Princeton |edition=Paperback |isbn=978-0-691-13626-4 | quote="...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." |url={{
|title=Topic: Provenance of a function
|author=John Conway
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The minimal distance of <math>x_0</math> to its ''i''-th lower and upper bounds equals
We define <math>\delta</math> as the minimum of all the finitely many <math>d_i.</math>
for all <math>i = 1,
▲for all <math>i = 1, ..., r,</math> <math>\quad |x_0 - k_i/i| \ge \delta\quad</math> and <math>\quad|x_0 - (k_i+1)/i| \ge \delta.</math>
This is to say, all these rational numbers <math> k_i/i,\;(k_i + 1)/i,\;</math> are outside the {{nowrap|<math>\delta</math>-neighborhood of <math>x_0.</math>}}
Now let <math>x \in \mathbb{Q} \cap (x_0 - \delta, x_0 + \delta)</math> with the unique representation <math>x = p/q</math> where <math> p, q \in \mathbb N</math> are coprime. Then, necessarily, <math> q > r,\;</math> and therefore,
Likewise, for all irrational <math> x \in I, \; f(x) = 0 = f(x_0),\;</math> and thus, if <math> \varepsilon > 0 </math> then any choice of (sufficiently small) <math>\delta > 0</math> gives
Therefore, <math>f</math> is continuous on <math> \mathbb R \setminus \mathbb Q.</math>
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|<math>f</math> has a strict '''[[maxima and minima|local maximum]]''' at each rational number.{{citation needed|date=September 2017}}
{{pb}}
|<math>f</math> is '''[[Riemann integrable]]''' on any interval and the integral evaluates to <math>0</math> over any set.
{{pb}}
|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>
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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.<ref name="Trifonov" />
==The ruler function==
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* [[Volterra's function]]
==
{{reflist}}
{{refbegin}}
*{{citation|last=Thomae |first=J. |year=1875 |title=Einleitung in die Theorie der bestimmten Integrale |edition= |publisher=Verlag von Louis Nebert |___location=Halle a/S |language=german}} <!-- author name as it appears in the (scanned) book -->
*{{citation|last=Abbott |first=Stephen |year=2016 |title=Understanding Analysis |edition=Softcover reprint of the original 2nd |publisher=[[Springer Science+Business Media|Springer]] |___location=New York |isbn=978-1-4939-5026-3}}
*{{citation |last1=Bartle |first1=Robert G. |last2=Sherbert |first2=Donald R. |year=1999 |title=Introduction to Real Analysis |edition=3rd |publisher=Wiley |isbn=978-0-471-32148-4 |url-access=registration |url=https://archive.org/details/introductiontore00bart_1 }} (Example 5.1.6 (h))
{{refend}}
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