On dense-lineability of sets of functions on RR

A subset MM of a topological vector space XX is said to be dense-lineable in XX if there exists an infinite dimensional linear manifold in M∪{0}M∪{0} and dense in XX. We give sufficient conditions for a lineable set to be dense-lineable, and we apply them to prove the dense-lineability of several subsets of C[a,b]C[a,b]. We also develop some techniques to show that the set of differentiable nowhere monotone functions is dense-lineable in C[a,b]C[a,b]. Other results related to density and dense-lineability of sets in Banach spaces are also presented.