Real Analysis Exchange

On the level structure of bounded derivatives.

F. S. Cater

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Abstract

We prove: In the space ${\mathcal C}$ of continuous functions on $[0,1]$ under the $sup$ metric, the functions all of whose level sets (in every direction) have measure zero, form a residual subset of ${\mathcal C}$. In the space ${\mathcal D}$ of bounded derivatives of $[0,1]$, the derivatives all of whose level sets are nowhere dense sets of measure zero form a residual subset of ${\mathcal D}$. Moreover, there exists a derivative in ${\mathcal D}$ all of whose level sets have measure zero and one of whose level sets is dense in $[0,1]$.

Article information

Source
Real Anal. Exchange, Volume 29, Number 2 (2003), 657-662.

Dates
First available in Project Euclid: 7 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149698556

Mathematical Reviews number (MathSciNet)
MR2083804

Zentralblatt MATH identifier
1064.26003

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A12: Rate of growth of functions, orders of infinity, slowly varying functions [See also 26A48]

Keywords
derivative level set measure category graph.

Citation

Cater, F. S. On the level structure of bounded derivatives. Real Anal. Exchange 29 (2003), no. 2, 657--662. https://projecteuclid.org/euclid.rae/1149698556


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References

  • A. M. Bruckner, K. M. Garg, The level structure of a residual set of continuous functions, Trans. Amer. Math. Soc., 232 (1977), 307–321.
  • F. S. Cater, A derivative often zero and discontinuous, Real Analysis Exchange, 11, no. 1 (1985/6), 265–270.
  • K. M. Garg, On a residual set of continuous functions, Czechoslovak Math. J., 20 (1970), no. 95, 537–543.
  • C. Goffman, Real Functions, Holt-Rinehart-Winston, New York, 1964.
  • Y. Katznelson, K. Stromberg, Everywhere differentiable, nowhere monotone functions, Amer. Math. Monthly, 81 (1974), 349–354.
  • C. Weil, The space of bounded derivatives, Real Analysis Exchange, 3 (1977/8), 38–41.
  • C. Weil, On nowhere monotone functions, Proc. Amer. Math. Soc., 56 (1976), 388–389.