I'm given $\gamma:(-\pi,\pi)\times \mathbb{R}\to S$ where $S$ is the one-sheet hyperbolid definied by the equation: $$x^2+y^2-z^2=1$$
Such that $\gamma(u,v)=(\cos(u)-v\sin(u),\sin(u)+v\cos(u),v)$.
I'm trying to show that $\gamma$ is a parametrization, I've already shown that $\gamma$ is smooth and its differential map is injective. I'm left to show that $\gamma^{-1}$ is continuous but I can't explicitly calculate it.
I've done some arithmetic and found two candidates which are $(x,y,z)\mapsto (\arctan(\frac{y-zx}{x+zy}),z)$ and $(x,y,z)\mapsto (\arccos(\frac{x+zy}{1+z^2}),z)$ but they wouldn't have the right codomains since $\arctan(\alpha)\in (-\frac{\pi}{2},\frac{\pi}{2})$ and $\arccos(\alpha)\in (0,\pi)$.
Is there a way to show continuity of $\gamma^{-1}$ without explicitly calculating it? I've thought using the inverse function theorem but the differential map at a point $p$ is given by a $3\times2$ matrix so how can it be a local isomorphism?
Moreover I think the parametrization would be missing the point $(-1,0,0)$ I can fix this by adding another parametrization but if we add $\pi$ this wouldn't happen, we can't do this since $(-\pi,\pi]$ isn't open in the usual topology. Is there a smarter way to fix the parametrization without adding one?