On dense-lineability of sets of functions on R

A subset $M$ of a topological vector space $X$ is said to be dense-lineable in $X$ if there exists an infinite dimensional linear manifold in $M\cup \left\{0\right\}$ and dense in $X$. 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 $C[a,b]$. We also develop some techniques to show that the set of differentiable nowhere monotone functions is dense-lineable in $C[a,b]$. Other results related to density and dense-lineability of sets in Banach spaces are also presented.