I have been looking the implicit function theorem's statement in various websites to solve the equation $$F(a,b)=0,\quad F:\mathbb{R}^{n+1}_+\rightarrow\mathbb{R},\quad a\in\mathbb{R}^{n}_+,b\in\mathbb{R}_+,\quad \mathbb{R}_+=[0,+\infty)$$ and most of them mention that in order to apply the implicit function theorem, I need an open neighborhood $U$ of $(a,b)$ such that $F(x,y)$ is continuously differentiable on $U$.
So my questions are the following:
- Does the neighborhood $U$ of $(a,b)$ need to be open? or maybe this condition can be relaxed? (for example, $U=A\times B$ where only $B$ needs to be open)
- If the neighborhood $U$ of $(a,b)$ needs to be open, does that mean that I can't apply the implicit function theorem on the boundary points of $\mathbb{R}^{n+1}_+$? because if $(a,b)$ is a boundary point of $\mathbb{R}^{n+1}_+$, then, there is no open neighborhood $U$ of $(a,b)$ where $F(x,y)$ is continuously differentiable since there would be points $(x,y)\in U$ but $(x,y)\notin\mathbb{R}^{n+1}_+$ where $F(x,y)$ is not defined (or maybe it is defined but not continuously differentiable).