Theorem Addressing inverse functions for $f : \mathbb{R}^m \to \mathbb{R}^n$ when $m > n$, $m < n$, and $m = n$

Question

I am looking for a theorem that talks about how a continuous function, or perhaps more specifically a polynomial, $f : A \to B$, where $\mathbb{R}^{m-1} \subset A \subseteq \mathbb{R}^m$ and $\mathbb{R}^{n-1} \subset B \subseteq \mathbb{R}^n$, can have/has to have/cannot have an inverse defined on it's codomain. Intuitively, if say $m=2$ and $n=1$, it make since that it can't have an inverse as most all inputs lead to level set outputs. Likewise, I have intuition for when inverse might exist if, say $m=1$ and $n=1$. However, I know there's likely some theory of real/complex analysis or topology that talks what conditions need to be met regarding the size of the domain set and that of the codomain.