If $f\in \mathcal{C}^1(\mathbb{R}^2)$, then there exists $g:[0,1]\to \mathbb{R}^2$ such that $f\circ g $ is constant.
Show there exists $g:(0,1)\to\mathbb{R}^2$ such that $h\circ g$ constant, $g$ continuously differentiable and $g$ injective