currently, I am looking for the existence of zeros of a vector valued function $\vec{f}(\vec{x}): \mathbb{R}^n \rightarrow \mathbb{R}^m$. For my specific problem, $n=2$ and $m=6$ (The dimensions could change as I make changes to function $f$). I know that I can expect a continuity of the function, but not continuous derivability due to the underlying physics.
I saw that there is a theorem called Poincaré–Miranda or Intermediate Value Theorem, which basically tells me that If I have $\vec{f}(\vec{x}_1) \leq \vec{0} \leq \vec{f}(\vec{x}_2)$ that there is a $\vec{f}(\vec{x_3}) = \vec{0}$. From the related Wikpedia entry it seems that this only holds for projections to the same dimensional space $\mathbb{R}^n \rightarrow \mathbb{R}^n$.
Question 1
Is there an extension of Poincaré–Miranda Theorem to functions $\vec{f}(\vec{x}): \mathbb{R}^n \rightarrow \mathbb{R}^m$?
Question 2
What if I artificially make my projection fulfill the requirements for the Poincaré–Miranda Theorem?
E.g.: I have a function $\vec{f}(\vec{x}): (x_1,x_2,x_3) \rightarrow (f_1(\vec{x}), f_2(\vec{x}))$. I know that both $f_1$ and $f_2$ are smooth. Then $f_3(\vec{x}) = f_1(\vec{x})+f_2(\vec{x})$ is also smooth. Also for $f_1(\vec{x_a}) \geq 0$ and $f_2(\vec{x_a}) \geq 0$ it follows that $f_3(\vec{x_a}) \geq 0$ and vice versa for $f_1(\vec{x_b}) \leq 0$ and $f_2(\vec{x_b}) \leq 0$. Hence, the theorem should be applicable to the new function $\vec{f_{new}}(\vec{x}): (x_1,x_2,x_3) \rightarrow (f_1(\vec{x}), f_2(\vec{x}), f_3(\vec{x}))$. Am I missing something obvious here?
Thanks :)