Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^1$ function. Prove that the restriction is not injective. Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^1$ function. And let $D$ an open subset of $\mathbb{R}^2$. Prove that the restriction of $f$ to $D$ is not injective.
Im trying to solve this but i dont know how...
The problem have a hint: Use the inverse theorem applied to an appropiate transformation in $\mathbb{R}^2$. 

Assume by contradiction that $f_D$ is injective. 
Let $U \subset D$ be an open connected subset of $D$. Then $f|_U$ is also injective. As $U$ is connected then $f(U)$ is connected, hence an interval. Let $d \in f(U)$ be an interior point.
As $f$ is injective, there exists an unique $e \in U$ so that $f(e)=d$.
Then $f$ is a continuous bijection between the connected set $U \backslash \{e \}$ and the disconnected set $f(U)\backslash \{ d \}$. 
