Let $f:\mathbb{R}^{2}\to \mathbb{R}$ be a $C^{1}$ function. Show that connected components of $\{(x,y)\in \mathbb{R}^{2}|f(x,y)=0, \frac{\partial f}{\partial x}(x,y)\neq 0\}$ are diffeomorphic to real line.
My thoughts: First, the connected components are the sets that satisfies $\frac{\partial f}{\partial x}(x,y)> 0$ or $\frac{\partial f}{\partial x}(x,y)< 0$
Now, I want to show these sets are diffeomorphic to real line.
With this condition, I can apply implicit function theorem. Then, exist $g:I\to \mathbb{R}$ such that $g$ is $C^{1}$ and $f(g(y),y)=0$.
But, now I don't know what I can do.