The following is a thought that I am hoping is true. I haven't been able to prove it and cannot seem to find a reference either. Ideas and references are appreciated.
Suppose $F_i: \mathbb{R}^n \to \mathbb{R}^n$ are continuous injective maps and the determinant of the Jacobian of $F_i$ is positive on $\mathbb{R}^n$. Consider $L_t: \mathbb{R}^n \to \mathbb{R}^n$ defined as $ L_t= tF_0+(1-t)F_1$ for each $t \in [0,1]$. Prove (or disprove) that there exists a $c>0$ such that $L_t$ is injective for each $t \in [0,c)$.