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Okamoto's one-parameter family of self-affine functions , where , includes the continuous nowhere differentiable functions of Perkins () and Bourbaki/Katsuura (), as well as the Cantor function (). The main purpose of this article is to characterize the set of points at which has an infinite derivative. We compute the Hausdorff dimension of this set for the case , and estimate it for . For all , we determine the Hausdorff dimension of the sets of points where: (i) ; and (ii) has neither a finite nor an infinite derivative. The upper and lower densities of the digit in the ternary expansion of play an important role in the analysis, as does the theory of -expansions of real numbers.
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Pieter Allaart, The infinite derivatives of Okamoto's self-affine functions: an application of -expansions. J. Fractal Geom. 3 (2016), no. 1 pp. 1–31