If $f:\mathbb{R}^2\to\mathbb{R}^1$ is of class $C^1$, show that $f$ is not one-to-one. If $f:\mathbb{R}^2\to\mathbb{R}^1$ is of class $C^1$, show that $f$ is not one-to-one. [Hint: If $Df(x) = 0$ for all $x$, then $f$ is constant. If $Df(x_0)\neq0$, apply the implicit function theorem.] 
Clearly there are two cases, if $Df(x)=0$ for all $x\in \mathbb{R}^2$ then $f$ is constant and therefore can not be one to one.
If there is a $x_0\in\mathbb{R}^2$ such that $Df(x_0)\neq 0$ then how can I use the implicit function theorem knowing that in order to apply it I have to ensure that $f(x_0)=0$ but this is not given in the problem? Here is the version of the theorem of the implicit function that I am using, thank you very much.


It suffices to prove the theorem in the case if $f(x_0) = 0$, because if you have an arbitrary injective $C^1$ map $f$ and $x_0$ where the $Df(x) \neq 0$, then you can apply the special case to $f(x)-f(x_0)$, which does satisfy the conditions for the implicit function theorem.
