On the boundary of $\Delta_\epsilon$ $f(x)$ will have at least one coordinate that is close to $A_i$

@PietroMajer My functions are essentially growthrates of species in biology, so you can assume that they do not behave very weird (however I never put mathematical terms to "very weird"). They are potentially $C^\infty$. I do however not really understand what you mean by uniformly close.

@PietroMajer They are assumed to be fixed. I guess uniformity of these limits would be inconsystent with 4. ( just intuition, without any deep thoughts).

Thanks, seems promising. I have however some questions: 1. There's a typo in the definition of $\mathbb{R}_0^n$, the sum should go over $x_i$. 2. I assume the usual combinations of proper functions are still proper, is this correct? 3. I only have to show, that $g$ is proper and that it's Jacobian vanishes nowhere, then the theorem is proven, correct? 4. Do you have any recommendation where I could read this up and possibly cite?

Not so sure about algebraic topology. I had a class of algebraic topology about 3 years ago or so. I thought that algebraic topology includes the extension of simply connectedness. My functions are not algebraic. In my commet to Meisman there is an example of $f$. 3. That might be a problem. Is it true, that the image of a injective function will be simply connected, when the preimage is simply connected? Because my $f$ can be made injectiv, by assuming $c_1$ = 1.

@MeisamSoleimaniMalekan The following functions proof, that 2 and 4 are not incompatible: $f_i (c) = \frac{c_i^n}{\Pi_j c_j}$

@MeisamSoleimaniMalekan The standard vectors are not allowed, as they contain non positive entries ($c\in\mathbb{R}^n_+$). Basically assumption 4 is why only positive vecotrs are allowed.