Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On differentiability properties of typical continuous functions and Haar null sets

Author: L. Zajícek
Journal: Proc. Amer. Math. Soc. 134 (2006), 1143-1151
MSC (2000): Primary 26A27; Secondary 28C20
Published electronically: September 28, 2005
MathSciNet review: 2196050
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ D$ ($ D^*$) be the set of all continuous functions $ f$ on $ [0,1]$ which have a derivative $ f'(x)\in \mathbf{R}$ ( $ f'(x)\in \mathbf{R}^*$, respectively) at least at one point $ x \in (0,1)$. B. R. Hunt (1994) proved that $ D$ is Haar null (in Christensen's sense) in $ C[0,1]$.

In the present article it is proved that neither $ D^*$ nor its complement is Haar null in $ C[0,1]$. Moreover, the same assertion holds if we consider the approximate derivative (or the ``strong'' preponderant derivative) instead of the ordinary derivative; these results are proved using a new result on typical (in the sense of category) continuous functions, which is of interest in its own right.

References [Enhancements On Off] (What's this?)

  • [BL] Y. Benyamini, J. Lindenstrauss, Geometric Nonlinear Functional Analysis, Vol. 1, Colloquium publications (American Mathematical Society); v. 48, Providence, Rhode Island, 2000. MR 1727673 (2001b:46001)
  • [D] R. Dougherty, Examples of non-shy sets, Fund. Math. 144 (1994), 73-88. MR 1271479 (96c:43001)
  • [G] K.M. Garg, Theory of Differentiation, John Wiley, New York, 1998. MR 1641563 (2000c:26002)
  • [H] B. R. Hunt, The prevalence of continuous nowhere differentiable functions, Proc. Amer. Math. Soc. 122 (1994), 711-717. MR 1260170 (95d:26009)
  • [HZ] P. Holický, L. Zajícek, Nondifferentiable functions, Haar null sets and Wiener measure, Acta Univ. Carolinae - Math. Phys. 41 (2000), 7-11. MR 1802330 (2001j:46058)
  • [J1] V. Jarník, Über die Differenzierbarkeit stetiger Funktionen, Fund. Math. 21 (1933), 48-58.
  • [J2] V. Jarník, Sur la dérivabilité des fonctions continues, Publications de la Fac. des Sc. de L'Univ. Charles 129 (1934), 9 pp.
  • [J3] V. Jarník, Sur la dérivée approximative unilatérale, Mem. Soc. R. Sci. Boheme, Cl. Sci. 9 (1934), 10 pp.
  • [K1] J. Kolár, Porous sets that are Haar null, and nowhere approximately differentiable functions, Proc. Amer. Math. Soc. 129 (2001), 1403-1408. MR 1814166 (2003d:28021)
  • [K2] J. Kolár, Private communication.
  • [LP] J. Lindenstrauss, D. Preiss, On Fréchet differentiability of Lipschitz maps between Banach spaces, Annals Math. 157 (2003), 257-288. MR 1954267 (2003k:46058)
  • [MZ] J. Malý, L. Zajícek, Approximate differentiation: Jarník points, Fund. Math. 140 (1991), 87-97. MR 1139090 (92m:26006)
  • [Sa] S. Saks, Theory of the integral, Monogr. Mat. VII, New York, 1937.
  • [Si] K. Simon, Some dual statements concerning Wiener measure and Baire category, Proc. Amer. Math. Soc. 106 (1989), 455-463. MR 0961409 (89k:26003)
  • [Z1] L. Zajícek, The differentiability structure of typical functions in $ C(0,1)$, Real Anal. Exchange 13 (1987/88), pp. 119, 113-116, 93.
  • [Z2] L. Zajícek, Differentiability properties of typical continuous functions, Real. Anal. Exchange 25 (1999/2000), 149-158.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 26A27, 28C20

Retrieve articles in all journals with MSC (2000): 26A27, 28C20

Additional Information

L. Zajícek
Affiliation: Charles University, Faculty of Mathematics and Physics, Sokolovská 83, 186 75 Prague 8, Czech Republic

Keywords: Typical continuous function, Haar null set, nowhere differentiable function, approximative derivative, preponderant derivative
Received by editor(s): March 5, 2004
Received by editor(s) in revised form: November 9, 2004
Published electronically: September 28, 2005
Additional Notes: This research was supported by MSM 113200007, GAČR 201/00/0767 and GAČR 201/03/0931
Communicated by: David Preiss
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society