Generating continuous mappings with Lipschitz mappings

If $X$ is a metric space, then $\mathcal{C}_{X}$ and $\mathcal{L}_X$ denote the semigroups of continuous and Lipschitz mappings, respectively, from $X$ to itself. The relative rank of $\mathcal{C}_{X}$ modulo $\mathcal{L}_{X}$ is the least cardinality of any set $U\setminus \mathcal{L}_{X}$ where $U$ generates $\mathcal{C}_{X}$. For a large class of separable metric spaces $X$ we prove that the relative rank of $\mathcal{C}_{X}$ modulo $\mathcal{L}_X$ is uncountable. When $X$ is the Baire space $\mathbb{N}^{\mathbb{N}}$, this rank is $\aleph_{1}$. A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.