Let $f:\mathbb{R}^2\to \mathbb{R}$ be a $C^1$ function. Then there exists a continuous one-to-one function $g$ on $[0,1]$ such that $f\circ g=$constsnt.
Attempt: If $f\equiv 0$ then nothing to prove. So assume that $f\neq 0$. That is there is some $(x_0,y_0)\in \mathbb{R}^2$ such that $f(x_0,y_0)\neq 0.$ Now we can apply Implicit function theorem to say that there is a function $\phi: U_{x_0}\to V_{y_0}$ such that $f(x,\phi(x))=0.$
Now how to get a function from $[0,1]$? This is not clear to me.
