Editing Weierstrass function (section)

==Construction==

[[File:Weierstrass Animation.gif|thumb|right|Animation based on the increasing of the b value from 0.1 to 5.]]

In Weierstrass's original paper, the function was defined as a [[Fourier series]]:

<math display="block">f(x)=\sum_{n=0} ^\infty a^n \cos(b^n \pi x),</math>

where <math display="inline">0<a<1</math>, <math display="inline">b</math> is a positive odd integer, and

<math display="block"> ab > 1+\frac{3}{2} \pi.</math>

The minimum value of <math display="inline">b</math> for which there exists <math display="inline">0<a<1</math> such that these constraints are satisfied is <math display="inline">b=7</math>. This construction, along with the proof that the function is not differentiable at any point, was first delivered by Weierstrass in a paper presented to the [[Prussian Academy of Sciences|Königliche Akademie der Wissenschaften]] on 18 July 1872.<ref>On [https://books.google.com/books?id=G-YaAAAAYAAJ&pg=PA560 page 560] of the 1872 {{lang|de|Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin}} (Monthly Reports of the Royal Prussian Academy of Science in Berlin), there is a brief mention that on 18 July, {{lang|de|"Hr. Weierstrass las über stetige Funktionen ohne bestimmte Differentialquotienten"}} (Mr. Weierstrass read [a paper] about continuous functions without definite [i.e., well-defined] derivatives [to members of the Academy]). However, Weierstrass's paper was not published in the {{lang|de|Monatsberichte}}.</ref><ref>Karl Weierstrass, [https://books.google.com/books?id=1FhtAAAAMAAJ&pg=PA71 {{lang|de|italic=no|"Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzeren einen bestimmten Differentialquotienten besitzen"}}] [On continuous functions of a real argument which possess a definite derivative for no value of the argument], in: {{lang|de|italic=no|Königlich Preußische Akademie der Wissenschaften}}, {{lang|de|italic=no|Mathematische Werke von Karl Weierstrass}} (Berlin, Germany:  Mayer & Mueller, 1895), vol. 2, pages 71–74.</ref><ref>See also: Karl Weierstrass, {{lang|de|Abhandlungen aus der Functionenlehre}} [''Treatises from the Theory of Functions''] (Berlin, Germany: Julius Springer, 1886), [https://books.google.com/books?id=fltYAAAAYAAJ&pg=PA97 page 97].</ref>

Despite being differentiable nowhere, the function is continuous: Since the terms of the infinite series which defines it are bounded by <math display="inline">\pm a^n</math> and this has finite sum for <math display="inline">0 < a < 1</math>, convergence of the sum of the terms is [[uniform convergence|uniform]] by the [[Weierstrass M-test]] with <math display="inline">M_n = a^n</math>. Since each partial sum is continuous, by the [[uniform limit theorem]], it follows that <math display="inline">f</math> is continuous. Additionally, since each partial sum is [[uniform continuity|uniformly continuous]], it follows that <math display="inline">f</math> is also uniformly continuous.

It might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be countably infinite or finite. According to Weierstrass in his paper, earlier mathematicians including [[Carl Friedrich Gauss|Gauss]] had often assumed that this was true. This might be because it is difficult to draw or visualise a continuous function whose set of nondifferentiable points is something other than a countable set of points. Analogous results for better behaved classes of continuous functions do exist, for example the [[Lipschitz functions]], whose set of non-differentiability points must be a [[Lebesgue null set]] ([[Rademacher's theorem]]). When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz or otherwise well-behaved. Moreover, the fact that the set of non-differentiability points for a [[Monotonic function|monotone function]] is [[measure zero|measure-zero]] implies that the rapid oscillations of Weierstrass' function are necessary to ensure that it is nowhere-differentiable.

The Weierstrass function was one of the first [[fractals]] studied, although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line. Rather between any two points no matter how close, the function will not be monotone.

The computation of the [[Hausdorff dimension]] <math display="inline">D</math> of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed that <math display="inline">D = 2 + \log_b(a) < 2</math>.<ref>Kenneth Falconer, ''The Geometry of Fractal Sets'' (Cambridge, England:  Cambridge University Press, 1985), pages 114, 149.</ref><ref>See also: Brian R. Hunt (1998), [http://www.ams.org/journals/proc/1998-126-03/S0002-9939-98-04387-1/S0002-9939-98-04387-1.pdf "The Hausdorff dimension of graphs of Weierstrass functions"], ''Proceedings of the American Mathematical Society'', vol. 126, no. 3, pages 791–800.</ref> That ''D'' is strictly less than 2 follows from the conditions on <math display="inline">a</math> and <math display="inline">b</math> from above. Only after more than 30 years was this proved rigorously.<ref>{{cite journal |author=Shen Weixiao |authorlink=Shen Weixiao |year=2018 |title=Hausdorff dimension of the graphs of the classical Weierstrass functions |journal=Mathematische Zeitschrift |issn=0025-5874 |doi=10.1007/s00209-017-1949-1 |arxiv=1505.03986 |s2cid=118844077 |volume=289 |issue=1–2 |pages=223–266}}</ref>

The term Weierstrass function is often used in [[real analysis]] to refer to any function with similar properties and construction to Weierstrass's original example. For example, the cosine function can be replaced in the infinite series by a [[Triangle wave|piecewise linear "zigzag" function]]. [[G. H. Hardy]] showed that the function of the above construction is nowhere differentiable with the assumptions <math display="inline">0 < a < 1, ab \geq 1</math>.<ref name="Hardy">Hardy G. H. (1916) "Weierstrass's nondifferentiable function", ''Transactions of the American Mathematical Society'', vol. 17, pages 301–325.</ref>