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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==Properties==

{{unordered list

|Thomae's function <math>f</math> is [[Bounded function|'''bounded''']] and maps all real numbers to the [[unit interval]]:<math>\;f : \mathbb R\; \to \;[0,\; 1].</math>

|<math>f</math> is [[periodic function|'''periodic''']] with period <math>1:\; f(x + n) = f(x)</math> for all [[integer]]s {{mvar|n}} and all real {{mvar|x}}.

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{{Collapse bottom}}

|<math>f</math> is [[Continuous function|'''discontinuous''']] at allevery rational numbersnumber, so its points of discontinuity are [[Dense orderset|dense]] within the real numbers.

{{Collapse top|title=Proof of discontinuity at rational numbers|width=80%}}

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|<math>f</math> is '''continuous''' at allevery [[irrational number]]s, alsoso its points of continuity are [[Dense set|dense]] within the real numbers.

{{Collapse top|title=Proof of continuity at irrational arguments|width=80%}}

Since <math>f</math> is periodic with period <math>1</math> and <math>0 \in \Q,</math> it suffices to check all irrational points in <math>I=(0,\;1).\;</math> Assume now <math>\varepsilon > 0,\; i \in \N</math> and <math>x_0 \in I \setminus \Q.</math> According to the [[Archimedean property]] of the reals, there exists <math>r \in \N</math> with <math>1/r < \varepsilon ,</math> and there exist <math>\; k_i \in \N,</math> such that

for <math>i = 1, \ldots, r</math> we have <math>0 < \frac{k_i}{i} < x_0 < \frac{k_i +1}{i}.</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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==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]]