As we know, it was Weierstrass who gave the first (published) example, in 1872, of a function which is continuous but everywhere non-differentiable. However, in his paper "Über continuirliche Functionen eines reellen Arguments, die für keinen Werth des letzeren einen bestimmten Differentialquotienten besitzen" there is no figure of this function or an attempt to draw it. Hence, is it known who was first to draw it? Was its depiction possible only to computer graphics or were there attempts before?

  • $\begingroup$ From the title of this question I thought it was going to be asking who first wrote down the $\wp$ for the Weierstrass $\wp$-function. $\endgroup$
    – KCd
    Jan 22 '16 at 5:19

I looked through some of my stuff this morning and found the following publications that seem to be relevant to your question.

Moore [1] (1900) and Koch [2] (1904), [3] (1906) are the earliest publications I could find that have diagrams of somewhat accurate approximating curves to nowhere differentiable continuous functions. Until fairly recently (1960s and later), almost all publications on this topic have no figures, and the few that do have figures tend to have only very superficial figures such as absolute value graphs or step functions that are used in various ways to generate such functions. Lang [4] (1961) is the earliest publication I could find that attempts to show the "final state" of the construction of a nowhere differentiable continuous function. Hailpern [5] (1976) is the earliest publication I could find on this topic that gives computer generated graphs, the graphs being for high level approximating curves for such a function. Tall [6] (1982), Dubuc [7] (1989), Hata [8] (1991), Duistermaat [9] (1992) also give various computer generated graphs, but none yet show the Weierstrass function. Baouche/Dubuc [10] (1992) is the earliest paper I could find with the Weierstrass function shown.

[1] Eliakim Hastings Moore, On certain crinkly curves, Transactions of the American Mathematical Society 1 (January 1900), 72-90.

See the various approximations to the coordinate functions of Peano/Hilbert's square-filling curves on pp. 81, 83.

[2] Helge von Koch, Sur une courbe continue sans tangente obtenue par une construction géométrique élémentaire, Arkiv för Matematik, Astronomi och Fysik 1 (1904), 681-702.

See Figure 4 on p. 698. Incidentally, Section III (pp. 697-702) of the paper involves a modification of the now-called Koch curve to obtain the graph of a (nowhere differentiable continuous) function. Incidentally, in the title of Section III translates to: "Transformation of $P$ to a curve $P'$ where the ordinate (i.e. $y$-coordinate) is a uniform function (i.e. single-valued function) of the abscissa (i.e. $x$-coordinate)".

[3] Helge von Koch, Une méthode géométrique élémentaire pour l'étude de certaines questions de la théorie des courbes planes, Acta Mathematica 30 (1906), 145-174.

I believe this is a republication of Koch's 1904 paper. Figure 4 and Section III mentioned above are on p. 167 and pp. 166-174.

[4] Lester Henry Lange, Successive differentiability, Mathematics Magazine 34 #5 (May-June 1961), 275-279.

Figure II on p. 279 is "An attempt to picture the function $f$ so defined", where $f$ is a nowhere differentiable continuous function given in Olmsted's 1956 book Intermediate Analysis.

[5] Brent Hailpern, Continuous non-differentiable functions, Pi Mu Epsilon Journal 6 #5 (1976), 249-260.

This paper has some computer generated graphs of approximations to Perkins' function (Amer. Math. Monthly 34, 1927, pp. 476-478) and Van der Waerden's function.

[6] David Tall, The blancmange function. Continuous everywhere but differentiable nowhere, Mathematical Gazette 66 #435 (March 1982), 11-22.

See Figure 1 on p. 11, Figure 5 on p. 14, Figure 6 on p. 15, Figure 7 on p. 16.

[7] Benoit Dubuc, On Takagi fractal surfaces, Canadian Mathematical Bulletin 32 #3 (September 1989), 377-384.

Figure 2.1 on p. 379 gives graphs of two Takagi-type functions.

[8] Masayoshi Hata, Topological aspects of self-similar sets and singular functions, pp. 255-276 in Jacques Bélair and Serge Dubuc (editors), Fractal Geometry and Analysis, Kluwer Academic Publishers [later published by Springer], 1991.

Proceedings of a 3-21 1989 conference in Montreal (Canada). Figure 3 on p. 271 shows the Besicovitch function (has at no point a one-sided finite or a one-sided infinite derivative). [Various piece-wise continuous approximations of this function can be found much earlier, but this appears to be a computer generated graph.] Figure 4 on p. 271 shows the Takagi function.

[9] J. J. Duistermaat, Selfsimilarity of 'Riemann's nondifferentiable function', Nieuw Archief voor Wiskunde (4) 9 (1991), 303-337.

Figure 1.1 on p. 304 gives a graph of Riemann's function. Figure 4.2 on p. 321 shows the behavior at the right side of a point where the derivative exists and is negative -- it looks like a choppier version of what $-x + x^2 \sin (x^{-2})$ looks like at the right side of $x=0.$ See also Figures 4.3, 4.4-4.5, 6.1, 6.2 on pp. 322, 323, 335, 336.

The following is from pp. 303-304: "Already for many years, a picture of the graph of $f(x)$ [the Riemann function], made by A.J. de Meijer, adorns the cover of the notes of the first semester analysis course in Utrecht. In it, the aforementioned differentiability properties at the rational points with not too large denominators can be distinguished quite clearly."

The following is from p. 308: "As for the role of the computer, of course in the old days computer pictures were not available to put one on the track. On the other hand, the mathematical analysis definitely is the more essential part of the story. Without it, one may look at the pictures with equal fascination, but with less understanding."

[10] Amar Baouche and Serge Dubuc, La non-dérivabilité de la fonction de Weierstrass, L'Enseignement Mathématique (2) 38 #1-2 (January-June 1992), 89-94.

Figure 1 on p. 90 gives a graph of the Weierstrass function.


I recently came across another paper that is old enough and relevant enough to belong to the above list (between [5] and [6]):

Michael Victor Berry and Zinaida V. Lewis, On the Weierstrass-Mandelbrot fractal function, Proceedings of the Royal Society A 370 #1743 (April 1980), 459-484.

This paper gives many computer generated graphs of a variation on the Weierstrass function that the authors call the Weierstrass-Mandelbrot function.


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