Note: The original question asked how to prove that $f$ maps interior points to interior points when it has a positive Jacobian. This was a silly question as it's an immediate consequence of the Inverse Function Theorem as pointed out in the comment. I modified the question to better reflect the true nature of the issue.
Consider a differentiable map $f$ with positive definite Jacobian: $$f(x,y):\Omega = [0,\infty)\times Y\longrightarrow\mathbb{R}^{m+1}, \text{ where } Y\subseteq\mathbb{R}^{m}\,\, \text{is compact, simply connected}.$$
My question is how to accurately describe the boundary of the image of $f.$ We know that from the Inverse Function Theorem that: $$F(\partial\Omega)\subseteq\partial F(\Omega).$$ But it seems that "new" boundary can form, if we look at the limit: $$g(y) = \lim\limits_{x\to\infty} f(x,y).$$ This limit exists and finite for any $y\in Y$ and for the particular problem I am working on, $g(y)$ turns out to be piecewise constant i.e.,: $$g(y) = \sum\limits_{s\in S}a_s\mathbf{1}_{\{X_s\}}(y)$$ where $Y = \cup_{s\in S}Y_s$ is a finite partition of $Y.$ So I am tempted to conclude: $$\partial F(\Omega) = F(\partial\Omega)\cup\{a_s\vert\ s\in S \}.$$
However, numerical simulation strongly suggests that the line segments (or hyperplanes) joining the points $a_s$ also lie on the boundary of image of $f.$ So at this point, I want to prove something like for two limit points $a_s,a_t$: $$\forall\alpha\in[0,1]: \alpha a_s + (1-\alpha)a_t\in\partial F(\Omega).$$
But none of the usual theorems about diffeomorphism I find is of direct help, as they deal with open sets or they are about holomorphic functions. However, my $\Omega$ is not open and my function is real.
Can someone point me to a nice reference that deals with results like this?