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 there exists an in any Lipschitz class of order less than one such that the set of knot points of has a measure .
On construit une famille de fonctions continues sur l’intervalle qui n’a nulle part de dérivée unilatérale finie ou infinie utilisant les équations fonctionnelles de De Rham. Puis on démontre que, pour tout , il existe une dans toute classe lipschitzienne d’ordre inférieur à 1 tel que la mesure de l’ensemble de nœud points de est égale à .
@article{AIF_1988__38_2_43_0, author = {Hata, Masayoshi}, 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}, doi = {10.5802/aif.1134}, mrnumber = {89i:26006}, zbl = {0641.26010}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1134/} }
TY - JOUR AU - Hata, Masayoshi TI - On continuous functions with no unilateral derivatives JO - Annales de l'Institut Fourier PY - 1988 SP - 43 EP - 62 VL - 38 IS - 2 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.1134/ DO - 10.5802/aif.1134 LA - en ID - AIF_1988__38_2_43_0 ER -
Hata, Masayoshi. On continuous functions with no unilateral derivatives. Annales de l'Institut Fourier, Volume 38 (1988) no. 2, pp. 43-62. doi : 10.5802/aif.1134. http://archive.numdam.org/articles/10.5802/aif.1134/
[1] Über die Baire'sche Ketegorie gewisser Funktionenmengen, Studia Math., 3 (1931), 174-179. | JFM | Zbl
,[2] Mémoire sur les nombres dérivés des fonctions continues, J. Math. Pures Appl. (Ser. 7), 1 (1915), 105-240. | JFM
,[3] On asymmetrical derivates of non-differentiable functions, Canad. J. Math., 20 (1968), 135-143. | MR | Zbl
,[4] On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), 381-414. | MR | Zbl
,[5] Über die Differenzierbarkeit stetiger Funktionen, Fund. Math., 21 (1933), 48-58. | JFM | Zbl
,[6] The Theory of Functions of a Real Variable, Toronto, 1951, pp. 172-181. | MR | Zbl
,[7] Sur les fonctions non dérivables, Studia Math., 3 (1931), 92-94. | JFM | Zbl
,[8] A continuous function with no unilateral derivatives, Trans. Amer. Math. Soc., 44 (1938), 496-507. | JFM | MR | Zbl
,[9] On continuous functions without a derivative, Fund. Math., 12 (1928), 244-253. | JFM
,[10] Sur quelques courbes définies par des équations fonctionnelles, Rend. Sem. Mat. Torino, 16 (1957), 101-113. | MR | Zbl
,[11] On the functions of Besicovitch in the space of continuous functions, Fund. Math., 19 (1932), 211-219. | JFM | Zbl
,[12] On functions without one-sided derivatives I, Proc. Benares Math. Soc., 3 (1941), 55-69. | MR | Zbl
,[13] On functions without one-sided derivatives II, Proc. Benares Math. Soc., 4 (1942), 95-108. | MR | Zbl
,[14] On the derivates of non-differentiable functions, Messenger of Math., 38 (1908), 65-69. | JFM
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