Thomae's function: Difference between revisions

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{{Short description|Function that is discontinuous at rationals and continuous at irrationals}}

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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">{{Citationcite journal |last1=Beanland |first1=Kevin |last2=Roberts |first2=James W. |last3=Stevenson |first3=Craig |date=2009 |title=Modifications of Thomae's Function and Differentiability |journal=[[The American Mathematical Monthly]] |volume=116 |issue=6 |pages=531–535 |jstor=40391145 |doi=10.4169/193009709x470425}}</ref>{{rp|p=531}}

<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''' (not to be confused with the integer [[ruler function]]),<ref>{{Citationcite book |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={{Google books|aYTYBQAAQBAJ|The Calculus Gallery|page=149|plainurl=yes}} | at = page 149, chapter 10}}</ref> the '''Riemann function''', or the '''Stars over Babylon''' ([[John Horton Conway]]'s name).<ref>{{cite web | url=http://mathforum.org/kb/message.jspa?messageID=1375516 | title=Topic: Provenance of a function | author=John Conway | publisher=The Math Forum | archiveurl=https://web.archive.org/web/20180613235037/mathforum.org/kb/message.jspa?messageID=1375516 | archivedate=13 June 2018}}</ref> Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration.<ref name="Thomae">{{citationcite book | 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 | at = p. 14, §20}} <!-- author name as it appears in the (scanned) book --></ref>

Since every [[rational number]] has a unique representation with [[coprime integers|coprime]] (also termed relatively prime) <math>p \in \mathbb Z</math> and <math>q \in \mathbb N</math>, the function is [[well-defined]]. Note that <math>q = +1</math> is the only number in <math>\mathbb N</math> that is coprime to <math>p = 0.</math>

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* For rational numbers, this follows from non-continuity.

* For irrational numbers:

*:For any [[sequence]] of irrational numbers <math>(a_n)_{n=1}^\infty</math> with <math>a_n \ne x_0</math> for all <math>n \in \mathbb{N}_{+}</math> that converges to the irrational point <math>x_0,\;</math>, the sequence <math>(f(a_n))_{n=1}^\infty</math> is identically <math>0,\;</math>, and so <math>\lim_{n \to \infty}\left|\frac{f(a_n)-f(x_0)}{a_n - x_0}\right| = 0.</math>.

*:According toOn [[Hurwitz'sthe theoremother (number theory)|Hurwitz's theorem]]hand, thereconsider also exists athe sequence of rational numbers <math>(b_n)_{n=1}^{\infty}</math> =with <math>b_n (k_n/n)_{n=1}^ \infty,lfloor nx_0\;rfloor/n</math>, converging towhere <math>x_0,\;lfloor nx_0\rfloor</math> withdenotes the [[Floor and ceiling functions|floor]] of <math>k_nnx_0</math>. Since <math>nx_0-1<\inlfloor nx_0\rfloor\mathbble Znx_0</math>, andthe sequence <math>(b_n)_{n =1}^{\ininfty}</math> \mathbbconverges Nto <math>x_0</math> coprimeusing andthe [[Squeeze theorem]]. Also, <math>|k_nb_n-x_0| = |\lfloor nx_0\rfloor/n - x_0| <= |\frac{1}{lfloor nx_0\sqrt{5}\cdotrfloor - nx_0|/n^2}. \;le 1/n</math> for all <math>n</math>.

*: Thus for all <math>n,</math>, <math>\left| \frac{f(b_n)-f(x_0)}{b_n - x_0} \right| >\ge \frac{1/n - 0}{1/(n} = 1</math>. Therefore we obtain <math>\sqrtliminf_{5n\to\infty} \cdotleft| n^2\frac{f(b_n)-f(x_0)} =\sqrt{5b_n-x_0} \cdotright| n\ge 1 \ne 0\;</math> and so {{nowrap|<math>f</math> is not differentiable}} at allany irrational number <math>x_0.</math>.

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|<math>f</math> has a strictproper '''[[maxima and minima|local maximum]]''' at each rational number, providing an example of a function with a dense set of proper local maxima.<ref>{{citationcite neededjournal|title=Solution to Problem 1129|first=Paolo|last=Perfetti|department=Problem Department|journal=Pi Mu Epsilon Journal|volume=12|issue=5|date=SeptemberFall 20172006|pages=301–319|jstor=24337958}} Perfetti supplies the negation of Thomae's function as an example with a dense set of proper local minima.</ref>

{{pb}}

See the proofs for continuity and discontinuity above for the construction of appropriate [[neighborhood (mathematics)|neighbourhoods]], {{nowrap|where <math>f</math> has}} maxima.

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|<math>f</math> is '''[[Riemann integrable]]''' on any interval and the integral evaluates to <math>0</math> over any set.

{{pb}}

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>{{citationcite book | last = Spivak | first = M. | author-link = Michael Spivak |year=1965 |title=Calculus on manifolds |publisher=Perseus Books | isbn = 978-0-8053-9021-6 | at = page 53, 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>

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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 [[Fσ set|{{math|''F''_σ}} set]]. If such a function existed, then the irrationals would be an {{math|''F''_σ}} set. The irrationals would then be the [[countable set|countable]] [[union (set theory)|union]] of [[closed set]]s <math display="inline">\bigcup_{i = 0}^\infty C_i</math>, but since the irrationals do not contain an interval, neither can any of the <math>C_i</math>. Therefore, each of the <math>C_i</math> 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.

Thomae’s function is also closely related to random processes generated by events within discrete uniform distributions such as pi digits, live roulette spins and dice rolls. A random variable C{{sub|i}} is repeatedly sampled N times from a discrete uniform distribution, where i ranges from 1 to N. For instance, consider integer values ranging from 1 to 10 or 1 to 6 in the case of a fair dice. Moments of occurrence, T{{sub|k}}, signify when events C{{sub|i}} repeat, defined as C{{sub|i}} = C{{sub|i-1}} or C{{sub|i}} = C{{sub|i-2}}, where k ranges from 1 to M, with M being less than N. Subsequently, define S{{sub|j}} as the interval between successive T{{sub|k}}, representing the waiting time for an event to occur. The random variable S is geometrically distributed. Finally, introduce Z{{sub|l}} as ln(S{{sub|j}}) – ln(S{{sub|j-1}}), where l ranges from 1 to U-1. The random variable Z displays fractal properties, resembling the shape distribution akin to Thomae's or Dirichlet function.<ref>Endrit Dheskali. Generation of a recurrent fractal process using discrete uniform distributions, 26 February 2024, PREPRINT (Version 1) available at Research Square [https://doi.org/10.21203/rs.3.rs-3987858/v1]</ref>

[[File:Thomae's function like distribution.jpg|Probability density function of the random variable Z]]

A variant of Thomae's function can be used to show that any {{math|''F''_σ}} subset of the real numbers can be the set of discontinuities of a function. If <math display="inline"> A = \bigcup_{n=1}^{\infty} F_n</math> is a countable union of closed sets <math> F_n</math>, define

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==References==

{{reflist}}

==Further reading==

{{refbegin}}

*{{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}}

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[[Category:Calculus]]

[[Category:Eponymous functions]]

[[Category:Fractals]]

[[Category:General topology]]

[[Category:Special functions]]