Hidden Harmony—Geometric Fantasies by Umberto Bottazzini & Jeremy Gray

Author:Umberto Bottazzini & Jeremy Gray
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(6.61)

where a is an odd integer and 0 < b < 1. Weierstrass limited himself to stating that f(x) is continuous.65 In order to prove that it is not differentiable at any point x 0 he observed that one can determine an integer α m (m an arbitrary positive integer) such that . By putting

one has x ′ < x 0 < x ′, such that x ′, x ′ approach x 0 arbitrarily closely for increasing m.

Under the further assumption that a straightforward calculation showed that the ratios

have opposite signs and become infinite as m → ∞. Thus, Weierstrass concluded, at x 0 the function f(x) has neither a finite nor an infinite, determined differential quotient.

However important this counterexample may be in the history of modern analysis, Weierstrass apparently did not attach too much significance to it. Indeed, he did not rush to publish his “piquant” example—as he used to call it—but let the paper be published only in vol. 2 of his Werke (Weierstrass 1872).

The first example of such functions appeared in print almost one year later, in September 1873, when Schwarz published his own example of a continuous nowhere differentiable function, which he had communicated one month earlier to the mathematical section of the Schweizerischen Naturforschenden Gesellschaft (Schwarz 1973). There Schwarz remarked that Riemann’s 1854 example of an integrable function discontinuous at every rational point, had in fact provided the example of a function—i.e. its primitive function—that is never differentiable at a rational point.

Eventually, Weierstrass’s counter-example appeared in print in 1875, when du Bois–Reymond published it and the relevant proof in his (1875, 29–31).66 He had received the example from Weierstrass who in turn communicated the paper to Borchardt for publication as early as November 1873. In his letters to du Bois–Reymond Weierstrass added that Riemann had once said that he had come across his function (6.60) in his research on elliptic functions, and commented at length on the fact that “now we know functions with properties of which one had no inkling before and which as you say, contradict all our previous ideas” (Weierstrass 1923a, 201).

Later on, on April 19, 1874 Weierstrass communicated to Schwarz an example found by Sonya Kovalevskaya adding that one could construct general series of the form having the same properties as his own function (6.61). However, he added, further research on this subject “would be of interest only if one could find general laws according to which one could be able to judge whether any of those functions is differentiable or not”. Assuming that a n and b n are positive constants, he tentatively conjectured that is nowhere differentiable if ∑ b n is convergent and divergent “but”, he concluded, “it does not seem easy to prove this rigorously”.

It is worth remarking here that one finds the differential calculus in almost every set of Weierstrass’s lecture notes on analytic function theory. However, the integral calculus is treated only in Killing’s 1868 notes, and in Weierstrass’s last lecture (see Sect. 6.9.4). In Killing’s notes, integrals are introduced at the end of the course,

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