I am curious to know the relationship between the set $Lip(X)$ of lipschitz functions and the all the sets $C(X), C_{0}(X), C^{k}(X), C_{c}(X), C^{\infty}(X), C_{0}^{\infty}(X)$ regarding density.
If you know where the set of Lipschitz functions is dense (and can reference) I'd appreciate very much. Also, I should mention that I am particularly curious about the cases $X = \mathbb{R}^n$ and $X=\Omega \subset \mathbb{R}^n$ where $\Omega$ is compact.
No matter what measure you put on the set of functions, the Lipschitz functions will always have measure 0 in the set continuous functions, therefore they are not dense. In fact, it can be proved that they are nowhere dense.