Suppose that we have a random vector $V\equiv (V_1,..., V_M)$ defined on the probability space $(\Omega, \mathcal{F}, P)$ with support $\mathcal{V} \subseteq \mathbb{R}^M$ that is an open subset of $\mathbb{R}^M$ with strictly positive Lebesgue measure. Moreover, $V$ has a distribution absolutely continuous with respect to Lebesgue measure on $\mathcal{V}$.
Question 1: Does this imply that the function $$ a\equiv (a_1,...,a_M)\in \mathbb{R}^M \mapsto H(a)\equiv \begin{pmatrix} P(V_1-V_k\geq a_k-a_1 \text{ }\forall k \neq 1)\\ ...\\ P(V_M-V_k\geq a_k-a_1 \text{ }\forall k \neq M)\\ \end{pmatrix}\in [0,1]^M $$ is continuous on $\mathbb{R}^M$?
Question 2: Let $\mathcal{V}=\mathbb{R}^M$. Does this imply that $H$ is continuous AND strictly increasing in each dimension?
My thoughts: I think that the answer to both is Yes because it comes from the properties of the cdf of $V$ that is continuous when $\mathcal{V}\subseteq \mathbb{R}^M$ and continuous and strictly increasing when $\mathcal{V}= \mathbb{R}^M$. But I still have doubts because we are dealing with differences between components of $V$.