On continuous functions with no unilateral derivatives
Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 43-62.

On construit une famille de fonctions continues sur l’intervalle [0,1] qui n’a nulle part de dérivée unilatérale finie ou infinie utilisant les équations fonctionnelles de De Rham. Puis on démontre que, pour tout α[0,1), il existe une f α dans toute classe lipschitzienne d’ordre inférieur à 1 tel que la mesure de l’ensemble de nœud points de f α est égale à α.

We construct a family of continuous functions on the unit interval which have nowhere a unilateral derivative finite or infinite by using De Rham’s functional equations. Then we show that for any α [0,1) there exists an f α in any Lipschitz class of order less than one such that the set of knot points of f α has a measure α.

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     title = {On continuous functions with no unilateral derivatives},
     journal = {Annales de l'Institut Fourier},
     pages = {43--62},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {38},
     number = {2},
     year = {1988},
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Hata, Masayoshi. On continuous functions with no unilateral derivatives. Annales de l'Institut Fourier, Tome 38 (1988) no. 2, pp. 43-62. doi : 10.5802/aif.1134. http://archive.numdam.org/articles/10.5802/aif.1134/

[1] S. Banach, Über die Baire'sche Ketegorie gewisser Funktionenmengen, Studia Math., 3 (1931), 174-179. | JFM | Zbl

[2] A. Denjoy, Mémoire sur les nombres dérivés des fonctions continues, J. Math. Pures Appl. (Ser. 7), 1 (1915), 105-240. | JFM

[3] K. M. Garg, On asymmetrical derivates of non-differentiable functions, Canad. J. Math., 20 (1968), 135-143. | MR | Zbl

[4] M. Hata, On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), 381-414. | MR | Zbl

[5] V. Jarnik, Über die Differenzierbarkeit stetiger Funktionen, Fund. Math., 21 (1933), 48-58. | JFM | Zbl

[6] R. L. Jeffery, The Theory of Functions of a Real Variable, Toronto, 1951, pp. 172-181. | MR | Zbl

[7] S. Mazurkiewicz, Sur les fonctions non dérivables, Studia Math., 3 (1931), 92-94. | JFM | Zbl

[8] A. P. Morse, A continuous function with no unilateral derivatives, Trans. Amer. Math. Soc., 44 (1938), 496-507. | JFM | MR | Zbl

[9] E. D. Pepper, On continuous functions without a derivative, Fund. Math., 12 (1928), 244-253. | JFM

[10] G. De Rham, Sur quelques courbes définies par des équations fonctionnelles, Rend. Sem. Mat. Torino, 16 (1957), 101-113. | MR | Zbl

[11] S. Saks, On the functions of Besicovitch in the space of continuous functions, Fund. Math., 19 (1932), 211-219. | JFM | Zbl

[12] A. N. Singh, On functions without one-sided derivatives I, Proc. Benares Math. Soc., 3 (1941), 55-69. | MR | Zbl

[13] A. N. Singh, On functions without one-sided derivatives II, Proc. Benares Math. Soc., 4 (1942), 95-108. | MR | Zbl

[14] W. H. Young, On the derivates of non-differentiable functions, Messenger of Math., 38 (1908), 65-69. | JFM

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