Consider a real analytic function $g: \mathbb{R}^m \rightarrow \mathfrak{M}_{\mathbb{R}}(n, k)$ to the set of $n \times k$ matrices where $k \ge n + 1$ such that $\forall x \in \mathbb{R}^m \quad \text{rank}(g(x)) \ge 1$ and for almost all points (i.e. except a set of zero measure) we also have $\text{rank}(g(x)) = n$, i.e. they are full-rank.
Let $h: \mathbb{R}^m \times \mathbb{R}^k \rightarrow \mathbb{R}^n$ be defined as $h(x, a) = g(x)a$ with the usual matrix-vector product. Is it true that the function $h$ is open wrt the usual topologies?
Also does the conclusion change if we add the requirements that the differential of $g$ is full-rank at almost every point?