On continuous functions with no unilateral derivatives

Masayoshi Hata

Annales de l'institut Fourier (1988)

  • Volume: 38, Issue: 2, page 43-62
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Hata, Masayoshi. "On continuous functions with no unilateral derivatives." Annales de l'institut Fourier 38.2 (1988): 43-62. <http://eudml.org/doc/74802>.

@article{Hata1988,
abstract = {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 $\alpha $$\in [0,1)$ there exists an $f_\{\alpha \}$ in any Lipschitz class of order less than one such that the set of knot points of $f_\{\alpha \}$ has a measure $\alpha $.},
author = {Hata, Masayoshi},
journal = {Annales de l'institut Fourier},
keywords = {continuous functions on the unit interval which have nowhere a unilateral derivative; de Rham's functional equations; set of knot points},
language = {eng},
number = {2},
pages = {43-62},
publisher = {Association des Annales de l'Institut Fourier},
title = {On continuous functions with no unilateral derivatives},
url = {http://eudml.org/doc/74802},
volume = {38},
year = {1988},
}

TY - JOUR
AU - Hata, Masayoshi
TI - On continuous functions with no unilateral derivatives
JO - Annales de l'institut Fourier
PY - 1988
PB - Association des Annales de l'Institut Fourier
VL - 38
IS - 2
SP - 43
EP - 62
AB - 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 $\alpha $$\in [0,1)$ there exists an $f_{\alpha }$ in any Lipschitz class of order less than one such that the set of knot points of $f_{\alpha }$ has a measure $\alpha $.
LA - eng
KW - continuous functions on the unit interval which have nowhere a unilateral derivative; de Rham's functional equations; set of knot points
UR - http://eudml.org/doc/74802
ER -

References

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  9. [9] E. D. PEPPER, On continuous functions without a derivative, Fund. Math., 12 (1928), 244-253. Zbl54.0275.02JFM54.0275.02
  10. [10] G. DE RHAM, Sur quelques courbes définies par des équations fonctionnelles, Rend. Sem. Mat. Torino, 16 (1957), 101-113. Zbl0079.16105MR20 #1733
  11. [11] S. SAKS, On the functions of Besicovitch in the space of continuous functions, Fund. Math., 19 (1932), 211-219. Zbl0005.39105JFM58.0256.03
  12. [12] A. N. SINGH, On functions without one-sided derivatives I, Proc. Benares Math. Soc., 3 (1941), 55-69. Zbl0063.07046MR5,175a
  13. [13] A. N. SINGH, On functions without one-sided derivatives II, Proc. Benares Math. Soc., 4 (1942), 95-108. Zbl0063.07047MR5,232e
  14. [14] W. H. YOUNG, On the derivates of non-differentiable functions, Messenger of Math., 38 (1908), 65-69. JFM39.0470.04

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