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On sparse subspaces of \(C[0,1]\). (English) Zbl 1104.26007

Summary: 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.

MSC:

26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
26A30 Singular functions, Cantor functions, functions with other special properties
54C05 Continuous maps
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