Let $U \subset \mathbb{R}^n$ be open and we are given a differentiable function $$\Psi: U \times \mathbb{R}^m \rightarrow \mathbb{R}^m, \quad (x, A) \mapsto \Psi(x,A)$$ with $m>n$.
We further assume that there is an open dense subset $V \subset U$ such that $$\forall x \in V \ \exists A \in \mathbb{R}^m: \quad \Psi(x,A)=0$$
By the implicit function theorem we know that $\forall x \in V$ exists an open neighbourhood $U_x \subset U$ where $A_{U_x}: U_x \rightarrow \mathbb{R}^m$ is a differentiable function and $\Psi(x,A_{U_x}(x))=0$ as long as the Jacobian at $x$ is surjective.
My question is the following: Can we "patch" all the functions $A_{U_x}$ together to get a differentiable function on U, $A:U \rightarrow \mathbb{R}^m $ such that in each open neighbourhood $U_x$ it holds $A|_{U_x}=A_{U_x}$?
Since $V$ is dense in $U$ I assume it is possible but I cannot come up with a proof.