Geodesic
The Subspace of $C[0,1]$ Consisting of Functions Having Finite One-Sided Derivatives Nowhere
Matematičeskie zametki, Tome 73 (2003) no. 3, pp. 348-354 
E. I. Berezhnoi. The Subspace of $C[0,1]$ Consisting of Functions Having Finite One-Sided Derivatives Nowhere. Matematičeskie zametki, Tome 73 (2003) no. 3, pp. 348-354. http://geodesic.mathdoc.fr/item/MZM_2003_73_3_a2/
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     title = {The {Subspace} of $C[0,1]$ {Consisting} of {Functions} {Having} {Finite} {One-Sided} {Derivatives} {Nowhere}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We construct a closed infinite-dimensional subspace $G\subset C[0,1]$giving an affirmative answer to the old question: Does there exist an infinite-dimensional closed subspace $G\subset C[0,1]$ such that each (not identically zero) function $y\in G$ has neither a right-hand nor a left-hand finite derivative at any point.

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