(I'm not sure the title describes the question adequately... edits are welcome)
Given a map $F(x,y)=(f(x,y), g(x,y))$,
such that, in some neighborhood of $(x_0, y_0)$, the following conditions hold:
- $f,g \in C^1$
- $|J_F(x,y)| \equiv 0 $
- $\frac{\partial f}{\partial x}(x,y) \ne 0 $
Prove that there exists a function $h$ which is continuously differntiable in some neighborhood of $f(x_0, y_0)$, such that $g(x,y)=h(f(x,y))$ in some neighborhood of $(x_0, y_0)$.
Guidance is to show that $g$ is independent of $y$ and depends only on x using an implicit function $x=\Phi(f(x,y), y)$ that exist in some neighborhood of $(x_0, y_0)$.
My attempts at a solution:
From the zero determinant of the Jacobian I can show that the derivatives of $g$ are linearly dependent on the derivatives of $f$ at every point.
Per the guidance, I can show that an implicit function $x=\Phi(f(x,y), y)$ exists (implicit function theorem conditions hold).
I'm stuck at this point... any help will be appreciated.