$f(u,v)=(ucosv,usinv,v)$, $0<u<1$, $0<v<2\pi$
Is $f$ homeomorphism between domain and image of $f$?
I noticed that function $g(x,y)=(\sqrt{x^2+y^2},z)$ is such that $g(f(u,v))=(u,v)$ but $f(g(x,y,z)) \neq (x,y,z)$
So $g$ isn't inverse of $f$, does $f$ have inverse?