I am reading Do Carmo's differential geometry book and the definition of a regular surface in the second chapter is given to be this:
I have few doubts about this definition:
1) Why we need to find a neighbourhood of point $p$? Is it because we can't always define a map $X$ that will work for the whole surface and we are trying to find local maps for every point. And later in chapter 3, when author talks about this parametrization/map for a surface, he says surface parametrized at point $p$, what does "at point $p$" means, are we talking about local parametrizations?
2) $X$ is differentiable to infinite order....is this really necessary? What if map $X$ is differentiable to a large but finite order?
3) X is continuous by condition 1. But I don't understand the use of continuity as described in the book. In book it is given that we need condition 2 for one to oneness of the map so that we can have single tangent plane at each point (basically to avoid self-intersections). So why don't just make condition 2 to be $X$ being one to one function?
4) I couldn't understand why we need condition 3 very clearly. Could you please give a geometric intuition for this?