Given a finite selection of random variables $X_1,X_2,\dots, X_n$ defined on a common background space $(\Omega, \mathcal{F}, P)$, when is the joint cdf $F_X:\mathbb{R}^n \to \mathbb{R}$ locally Lipschitz continuous with $X=(X_1,X_2,\dots, X_n)$ given as the bundling.
In the case $n=1$, we know that (global) Lipschitz implies (global) absolute continuity, hence the existence of a Lebesgue density for the distribution $X(P)$. I would assume that this statement may be copied to the "local" setting. Additionally, assuming $n=1$, $F_X$ is (everywhere) differentiable iff $X(P)$ admits a continuous (Lebesgue) density. Thus $F_X$ is differentiable and locally Lipschitz iff the (continuous) density of $X(P)$ is locally bounded.
Is it possible to generalize these properties to $n>1$? Or perhaps approach the initial question by other means?