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I wonder that whether there exist a nowhere differentable continuous function with its graph in $\mathbb{R}^2$ has Hausdorff dimension $1$.

A result about Weierstrass's function is that $\sum_{k=1}^{\infty}a^{(s-2)k}\cos a^kx$ has Hausdorff dimension $s$, provided $1<s<2$. But I don't know more about the case $s=1$.

cosecant
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1 Answers1

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The Takagi function (see [1]), rediscovered by Van der Waerden, is nowhere differentiable (see [1] or [2]). The Hausdorff dimension of its graph is 1 (see [3]). More information is in the survey [4].

[1] T. Takagi, A simple example of the continuous function without derivative, Phys.-Math. Soc. Japan 1 (1903), 176-177. The Collected Papers of Teiji Takagi, S. Kuroda, Ed., Iwanami (1973), 5–6.

[2] P. Billingsley, Van Der Waerden’s Continuous Nowhere Differentiable Function, Amer. Math. Monthly 89 (1982), no. 9, 691.

[3] R. Mauldin and S. Williams, On the Hausdorff dimension of some graphs, Trans. Amer. Math. Soc. 298 (1986), 793–803.

[4] Allaart, Pieter C.; Kawamura, Kiko (11 October 2011), The Takagi function: a survey, arXiv:1110.1691

Yuval Peres
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  • FYI, the Takagi function has a two-sided derivative of $+\infty$ on a countable dense set and a two-sided derivative of $-\infty$ on another countable dense set, so one might wonder whether a continuous function that is nowhere finitely or infinitely differentiable can have Hausdorff dimension $1.$ In fact, such functions are pretty much everywhere in a certain sense -- most continuous functions in the Baire category sense (using sup norm) have Hausdorff dimension $1$ (see the Mauldin/Williams paper or this paper), and so using the (continued) – Dave L. Renfro Jun 04 '22 at 00:11
  • results mentioned in this answer, most continuous functions are nowhere finitely or infinitely differentiable (and much worse than this) while still having the minimal graph size for Hausdorff dimension. There remains the question of whether there exists a Besicovitch function (continuous having no finite or infinite one-sided derivative at any point) of Hausdorff dimension $1,$ since such functions only form a Baire-meager subset of the continuous functions. This is probably known, but I don't have time now to investigate further. – Dave L. Renfro Jun 04 '22 at 00:11