On Special Partitions of [0,1] and Lineability within Families of Bounded Variation Functions

We show that there exist large algebraic structures (vector spaces, algebras, closed subspaces, etc.) formed entirely (except for 0), on one hand, by singular, nowhere monotonic functions on [0,1] and, on the other hand, by absolutely continuous nowhere monotonic functions. Several tools, of independent interest, related to obtaining special partitions of

into uncountable collections will be provided and used. The results obtained in this note are either new or improved versions of already existing ones.