The Subspace of $C[0,1]$ Consisting of Functions Having Finite One-Sided Derivatives Nowhere

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.