Prove that the Inverse Parametric equation is continuous. Original question: 
Show that a cylinder {$(x,y,z) \in  R^3 : x^2+y^2=1$} is a regular surface, and 
find parametrization whose coordinate neighborhoods cover it. 
First of all, what do it mean to cover it? 
Secondly I let $X = (cos(t),sin(t),z)$ on $R^2 \to R^3$   for my parameterization and it is obviously differentiable in the sense that its partial derivatives of any variables and degree exist. 
Second it's continuous since it's differentiable. X is 1-1 since z is 1-1. 
How do we show X is onto? Also is both t and z my 2 variables in $R^2$ that I am using?
Also how do I prove that $X^{-1}$ is also continuous? 
Thirdly I have to show that for each point in a subset U of $R^2$ its differential is 1-1. The book says this is true if the Jacobian is non-zero. 

With your definition, $X$ is not itself 1-1 because $(t,z)$ and $(t+2\pi n,z)$ map to the same point on the cylinder. One has to restrict the $t$ to intervals somehow to avoid this, and include all the covered values, and this means there will be two (at least) parts or "charts" for the map.
