Banach spaces and Banach lattices of singular functions
The class of real singular functions on the unit interval, that is, those continuous bounded variation functions having null derivative almost everywhere, is studied from the point of view of lineability. In particular, large closed vector subspaces, large linear algebras and large Banach lattices are found to live, except for zero, inside several subclasses of it. These subclasses are related to the size of the zero set, to nowhere monotonicity, or to the existence of noncritical points. Also the family of continuous functions that are constant on full measure sequences of sets is analyzed from this point of view. Moreover, it is studied what happens in this context when one turns from bounded variation topology to uniform convergence topology.