Thomae's function: Difference between revisions

Revision as of 05:30, 20 December 2022

Revision as of 05:31, 20 December 2022

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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={{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">{{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 | at = p. 14, §20}} <!-- author name as it appears in the (scanned) book --></ref>

|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>{{Harvnb|Thomae|1875|p=14|loc=§20}}</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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{{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))