Real Analysis Exchange

On sparse subspaces of C[0,1].

F. S. Cater

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Abstract

We prove the existence of three subspaces of $C[0,1]$, each homeomorphic to $C[0,1]$; the first consists only of infinitely many times differentiable functions, the second consists only of singular functions of bounded variation, and the third consists only of nowhere differentiable functions.

Article information

Source
Real Anal. Exchange, Volume 31, Number 1 (2005), 7-12.

Dates
First available in Project Euclid: 5 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149516800

Mathematical Reviews number (MathSciNet)
MR2218184

Zentralblatt MATH identifier
1104.26007

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A27: Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives 26A30: Singular functions, Cantor functions, functions with other special properties 54C05: Continuous maps

Keywords
homeomorphism differentiable functions singular functions.

Citation

Cater, F. S. On sparse subspaces of C[0,1]. Real Anal. Exchange 31 (2005), no. 1, 7--12. https://projecteuclid.org/euclid.rae/1149516800


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References

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  • A. P. Morse, A Continuous Function With No Unilateral Derivatives, Trans. Amer. Math. Soc., 44 (1938), 496–507.