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.

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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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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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