Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define $$ K=\{x\in\mathbb{R}^2|f(x)=0\}. $$

I wish to know whether there is a continuously differentiable function $g:\mathbb{R}\to\mathbb{R}^2$ such that $$ \forall k\in K,\quad\exists u\in\mathbb{R}, \quad g(u)=k. $$ So it is like $K$ can be expressed as an image of a continuosly differentiable function $g$.

If $f$ is linear there of course we can find $g$. It is still true for any function? Any counter example?

I am thinking $f(x,y)=x^3+y^3-xy$. But, how would we prove it mathematically?