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.


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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