# Show that a differentiable function on a convex space is injective

Let $n \in \Bbb N$ and $G \subset \Bbb R^n$ be a convex space, $f: G \to \Bbb R^n$ continuously differetiable and

$$\det\begin{pmatrix} \frac{\partial f_1}{\partial x_1}(c_1)& \cdots &\frac{\partial f_1}{\partial x_n}(c_n) \\\vdots & \vdots & \vdots \\\frac{\partial f_n}{\partial x_1}(c_n)& \cdots & \frac{\partial f_n}{\partial x_n}(c_n)\end{pmatrix}\neq 0,\quad \forall\,c_1,c_2,\dots,c_n \in G.$$

Now I have to show that $f$ is injective. Any ideas or tipps on how to show this? Thanks in advance!

• Counterexample? mathoverflow.net/a/104925/76061 – A.Γ. Jul 2 '16 at 17:22
• @A.G. You did notice the comment to the answer you are pointing to? – Thomas Jul 2 '16 at 18:26
• @Thomas Yes, but the question there was different (assuming all principal minors are nonzero). – A.Γ. Jul 2 '16 at 18:32