Journal of Humanistic Mathematics Volume 7 | Issue 2 July 2017 On French Pudding and a German Mathematician Amy Shoemaker Stanford University Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Arts and Humanities Commons, and the Mathematics Commons Recommended Citation Shoemaker, A. "On French Pudding and a German Mathematician," Journal of Humanistic Mathematics, Volume 7 Issue 2 (July 2017), pages 239-258. DOI: 10.5642/jhummath.201702.11 . Available at: https://scholarship.claremont.edu/jhm/vol7/iss2/11 ©2017 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. On French Pudding and a German Mathematician Cover Page Footnote A special thanks to Professor Gizem Karaali for her guidance and instruction in my first foray into the world of Analysis back in 2011, and for all of her support and encouragement since then. This work is available in Journal of Humanistic Mathematics: https://scholarship.claremont.edu/jhm/vol7/iss2/11 On French Pudding and a German Mathematician Amy Shoemaker Institute for Computational & Mathematical Engineering, Stanford University, CA, USA [email protected] Synopsis At the turn of the 19th century, mathematics developed the rigor that is now considered essential to the field. Mathematicians began going back and proving theorems and statements that had been taken to be true on face value, ensuring the underpinnings of mathematics were solid. This era of mathematics was char- acterized not only by setting foundations, but also by pushing the boundaries of new ideas. In 1830, Bolzano found an example of a function that was nowhere differentiable, despite being continuous. Thirty years later, Cellerier and Riemann each discovered another example of such a pathological function. The first ev- erywhere continuous nowhere differentiable function to be published appeared in Borchardt's Journal in 1875. This function was proposed by Karl Weierstrass. His function, in addition to similar examples that followed it, revolutionized the ideas of continuity, differentiability, and limits. Due to these pathological func- tions, the traditional definitions were rethought and revised. Here we explore some of these pathological functions. 1. A Historical Introduction \Mathematics" has etymological roots in the Greek word \µαθαιν!", mean- ing \learn". Prior to the 19th century, the field was characterized by an intuitive investigation of numbers, quantity, and space. The new pieces of knowledge being discovered and learned were far from rigorous and occasion- ally even faulty. At the turn of the 19th century, though, Mathematics took a more critical approach to its namesake, to learning. It was Analysis that brought about this rigor by going back and proving theorems and statements that had been taken to be true on face value. Journal of Humanistic Mathematics Vol 7, No 2, July 2017 240 On French Pudding and a German Mathematician One such assumed truth was Amp´ere'sTheorem, which stated that every continuous function is differentiable except at a few isolated points. Indeed, functions like y = jxj, with problematic points of non-differentiability (in this case the corner at x = 0) were known to exist. The prospect of a function's being everywhere continuous yet nowhere differentiable, however, seemed absurd. At the close of the 19th century, in 1899, Poincar´esaid, \A hundred years ago, such a function would have been considered an outrage on common sense." Indeed, it would have been. [5] The 19th century, however, was an opportune environment for mathemati- cians to push the limits of their field (pun intended). In 1821, Cauchy pub- lished his famous textbook that contained requirements for the rigorization of mathematics, Cours d' Analyse [3]. The text begins with a list of questions (when we read between the lines): • What is a derivative really? Answer: a limit. • What is an integral really? Answer: a limit. • What is an infinite series a1 + a2 + a3 + ··· really? Answer: a limit. • This leads to: What is a limit? Answer: a number. • And, finally, the last question: What is a number? And thus mathematicians like Weirestrass, Heine, Cantor, and others began to rigorously define the foundations of analysis, and thereby, of mathematics [5]. This era of mathematics was not only characterized by a foundational fo- cus, but also by a pushing of the boundaries of new ideas. In 1830, Bolzano found an example of a function that was nowhere differentiable, despite be- ing continuous. Thirty years later, Cellerier and Riemann each discovered an example of such a pathological function. The first everywhere continu- ous nowhere differentiable function to be published appeared in Borchardt's Journal in 1875. This function was proposed by Karl Weierstrass. His func- tion, in addition to similar examples that superseded it, revolutionized the ideas of continuity, differentiability, and limits. Due to these pathological functions, the traditional definitions were rethought and revised. [5, 11] Amy Shoemaker 241 Prior to the revision, the accepted definitions of limits and continuity were those of Cauchy: \. f(x) will be called a continuous function, if . the numerical values of the difference f(x + α) − f(x) decrease indefinitely with those of α : : :"[3, page 43]. In 1874, Weierstrass and Bolzano brought the definition of limits to the precise definition we still currently use: \Here we call a quantity y a continuous function of x, if after choosing a quantity " the existence of δ can be proved, such that for any value between x0 −δ : : : x0 +δ the corresponding value of y lies between y0 − " : : : y0 + "." [5] Here we have very briefly summarized the history of what is today known as the birth of rigorous analysis. Readers interested in this history will find care- ful scholarship and much detail in [4]; a more recent exploration of Cauchy's introduction to [3] can be found in [1]. Our focus here is merely mathemat- ical. So without further ado, let us delve into the seemingly mind-bending, logic-defying functions that in fact are truth-disclosing, logical certitudes. 2. Defining Continuity and Differentiability Not too soon, though! Before delving into specific examples proving the exis- tence of everywhere continuous nowhere differentiable functions, it would be beneficial to review the definitions of continuity and differentiability. There- fore we begin with the basic definition below: Continuity A function f : D R is continuous at c if for every " > 0, there exists a δ > 0 such that for every x 2 D, if jx − cj < δ, then jf(x) − f(c)j < ". In the following we will also need the notion of uniform continuity: For a function to be uniformly continuous on some interval D, it must, for a single δ, satisfy the above definition of continuity at c for every point c 2 D. Next, we provide the definition of differentiability: Differentiability Let I be an interval in the domain, and let c 2 I. The function f : I R is differentiable at c if the following limit (denoted by f 0(c)) exists: f(x) − f(c) f 0(c) = lim x!c x − c For a function to be nowhere differentiable, the above limit must not exist at every point c. 242 On French Pudding and a German Mathematician 3. Weierstrass' Function (1875) 3.1. Historical Background In one of his lectures at the Royal Academy in Berlin in 1872, Weierstrass proposed the existence of everywhere continuous nowhere differentiable func- tions. He said, \As I know from some pupils of Riemann, he as the first one (around 1861 or earlier) suggested as a counterexample to Amp´ere's Theorem [which perhaps could be interpreted as: every continu- ous function is differentiable except at a few isolated points]; for example, the function R does not satisfy this theorem. Unfor- tunately, Riemanns proof was unpublished and, as I think, it is neither in his notes nor in oral transfers..." Weierstrass then went forth to disprove Amp´ere'sTheorem with a proof of the existence of continuous, nowhere differentiable functions. The function he used as an example was an infinite sum of cosine functions that formed a fractal. His lecture notes were given to Borchardt's Journal by du Bois- Reymond in 1873, but Weierstrass spent two years attempting to make more progress and remarks. In 1875, the proof was published in almost identical form. [11] 3.2. The Function The original Weierstrass function was: 1 X f(x) = bncos(anx); (1) n=1 3π for 0 < a < 1, b > 1 is an odd integer, and ab > 1 + 2 . The many restrictions placed on the variables were simplified by Hardy [6] in 1916, but the function retained all of its properties and general form (see Figure1 on the following page). Amy Shoemaker 243 Figure 1: Plot of the Weierstrass function. A section of the plot is zoomed in on to illustrate the fractal nature of the function. Public domain image created by Eeyore22 on 26 October 2008, available at https://en.wikipedia.org/wiki/Weierstrass_function, accessed last on June 17, 2017. 3.3. Proof of Uniform Continuity and Nowhere Differentiability Before proving the continuity of the function f, we must delve into some of Weierstrass's other findings. In particular, we will describe the Weierstrass M-test.