Real Analysis Exchange

On sparse subspaces of C[0,1].

F. S. Cater

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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

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

First available in Project Euclid: 5 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

homeomorphism differentiable functions singular functions.


Cater, F. S. On sparse subspaces of C[0,1]. Real Anal. Exchange 31 (2005), no. 1, 7--12.

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