In the book of Analysis on Manifolds by Munkres, at page 209 question 3, it is asked that
and I have given the following (partial) answer to it:
Solution:
Let $M \subseteq \mathbb{R}^3$ denote the set of solutions that that system of equations satisfies the given condition. Then
if $$det ( \frac{\partial (f,g)}{\partial (y,z)} ) \not = 0 \quad \forall x \in M,$$ by the implicit function theorem, there exists $a_y, a_z, b_y, b_z: \mathbb{R} \to \mathbb{R}$ of class $C^r$ s.t
$$f(x, a_y(x), a_z (x)) = 0 \quad and \quad g(x, b_y (x), b_z (x)) = \quad \forall x \in N,$$ where $N = \{ \pi_1 (x,y,z) \in \mathbb{R} | (x,y,z) \in M\}.$
Now define $\alpha : \mathbb{R} \to \mathbb{R}^3$ by
$$ \alpha (x) = (x, f(x, a_y(x), a_z (x)), g(x, b_y (x), b_z (x))) \quad \forall x \in N,$$
and observe that $D\alpha$ exists (because of our assumtion) and equal to
$$D \alpha = [1, Df, Dg]^T,$$ which is rank 1 for all $x \in N.$
Moreover, it is clear that $\alpha \in C^r$.
However, as it is in this question, I couldn't show the continuity of $\alpha^{-1}$.
Hence the question;
Question:
How can we show the continuity of $\alpha^{-1}$, and is the rest of the answer (assuming we have answered the first part) is correct ?