I've been trying to solve this exercise: If $f:\mathbb R^3\rightarrow \mathbb R^2$ is continuously differentiable, then $f$ is not injectve.
I was thinking in use the implicit function theorem, in the sense that if I can see that the matrix $$df(a)=\begin{bmatrix}\frac{df}{dx}&\frac{df}{dy} \end{bmatrix},$$ where $y=(y_1,y_2)$, has $\frac{df}{dy}$ non singular, it follows that exist a $g:\mathbb R\rightarrow \mathbb R$ continuously differentiable, and that $f(x,g(x))=c$ for all $x\in U\subset\mathbb R$, for some open set $U$, therefore $f$ is not injective.
The problem is that I do not know how to show the existence of such $a$.
Thanks in advance.