Editing Weierstrass function (section)

{{short description|Function that is continuous everywhere but differentiable nowhere}}
{{distinguish|text=the [[Weierstrass elliptic function]] (<math>\wp</math>) or the [[Weierstrass functions|Weierstrass sigma, zeta, or eta functions]]}}
{{Use dmy dates|date=September 2020}}
[[Image:WeierstrassFunction.svg|300px|thumb|Plot of Weierstrass function over the interval [&minus;2,&nbsp;2]. Like some other [[fractal]]s, the function exhibits [[self-similarity]]: every zoom (red circle) is similar to the global plot.]]

In [[mathematics]], the '''Weierstrass function''', named after its discoverer, [[Karl Weierstrass]], is an example of a real-valued [[function (mathematics)|function]] that is [[continuous function|continuous]] everywhere but [[Differentiable function|differentiable]] nowhere. It is also an example of a [[fractal curve]].

The Weierstrass function has historically served the role of a [[pathological (mathematics)|pathological]] function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points.<ref>At least two researchers formulated continuous, nowhere differentiable functions before Weierstrass, but their findings were not published in their lifetimes.

Around 1831, [[Bernard Bolzano]] (1781–1848), a Czech mathematician, philosopher, and Catholic priest, constructed such a function; however, it was not published until 1922. See:

* Martin Jašek (1922) [http://dml.cz/bitstream/handle/10338.dmlcz/121916/CasPestMatFys_051-1922-2_2.pdf "Funkce Bolzanova"] (Bolzano's function), ''Časopis pro Pěstování Matematiky a Fyziky'' (Journal for the Cultivation of Mathematics and Physics), vol. 51, no. 2, pages 69–76 (in Czech and German).
* [[Vojtěch Jarník]] (1922) "O funkci Bolzanově" (On Bolzano's function), ''Časopis pro Pěstování Matematiky a Fyziky'' (Journal for the Cultivation of Mathematics and Physics), vol. 51, no. 4, pages 248 - 264 (in Czech).  Available on-line in Czech at:  http://dml.cz/bitstream/handle/10338.dmlcz/109021/CasPestMatFys_051-1922-4_5.pdf  .  Available on-line in English at:  http://dml.cz/bitstream/handle/10338.dmlcz/400073/Bolzano_15-1981-1_6.pdf  .
* Karel Rychlík (1923) "Über eine Funktion aus Bolzanos handschriftlichem Nachlasse" (On a function from Bolzano's literary remains in manuscript), ''Sitzungsberichte der königlichen Böhmischen Gesellschaft der Wissenschaften'' (Prag) (Proceedings of the Royal Bohemian Society of Philosophy in Prague) (for the years 1921-1922), Class II, no. 4, pages 1-20.  (''Sitzungsberichte'' was continued as:  ''Věstník Královské české společnosti nauk, třída matematicko-přírodovědecká'' (Journal of the Royal Czech Society of Science, Mathematics and Natural Sciences Class).)

Around 1860, Charles Cellérier (1818 - 1889), a professor of mathematics, mechanics, astronomy, and physical geography at the University of Geneva, Switzerland, independently formulated a continuous, nowhere differentiable function that closely resembles Weierstrass's function.  Cellérier's discovery was, however, published posthumously:

* Cellérier, C. (1890) [https://books.google.com/books?id=HMghAQAAIAAJ&pg=PA142 "Note sur les principes fondamentaux de l'analyse"] (Note on the fundamental principles of analysis), ''Bulletin des sciences mathématiques'', second series, vol. 14, pages 142 - 160.</ref> Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of [[smoothness]]. These types of functions were disliked by contemporaries: [[Charles Hermite]], on finding that one class of function he was working on had such a property, described it as a "lamentable scourge"{{Disputed inline|Accuracy of Poincaré quote in lead|date=March 2025}}.<ref>{{cite book |last1=Hermite |first1=Charles |author-link1=Charles Hermite |last2=Stieltjes |first2=Thomas |author-link2=Thomas Joannes Stieltjes |title=Correspondance d'Hermite et de Stieltjes |chapter=Letter 374, 20 May 1893 |editor-last1=Baillaud |editor-first1=Benjamin |editor-last2=Bourget |editor-first2=Henri |volume=2 |publisher=Gauthier-Villars |year=1905 |pages=317–319 |language=fr}}
</ref> The functions were difficult to visualize until the arrival of computers in the next century, and the results did not gain wide acceptance until practical applications such as models of [[Brownian motion]] necessitated infinitely jagged functions (nowadays known as fractal curves).<ref>{{cite web | url = https://nautil.us/maths-beautiful-monsters-234859/ | title = Math's Beautiful Monsters: How a destructive idea paved the way for modern math | last = Kucharski | first = Adam | date = 2017-10-26 | access-date = 2023-10-11}}</ref>