I am trying to prove that $S^1 :=\{ (x,y) | x^2+y^2=1\}$ is a manifold. I take two open subsets:
$U_N := \{ S^1 \text{ north pole removed}\}$
$U_S := \{ S^1 \text{ south pole removed}\}$
Obviously, they cover $S^1$.
I define two maps $f_N$ and $f_S$ on $U_N$ and $U_S$ by steographic projection onto $\Bbb R$. (hoping that they are homeomorphisms)
My ultimate goal was to prove that both $f_N \ \circ f_S^{-1}$ and $f_S \ \circ f_N^{-1}$ are differentiable on $f_S(U_N \cap U_S)$ and $f_N(U_N \cap U_S)$, respectively.
This would prove that $S^1$ is a smooth manifold. (Question 1: Am I right?)
Question 2: To prove homeomorphism, $f_N$ and $f_S$ should be continuous. However, the points on the left and on the right of the pole are mapped to very far away. So, they are not continuous.
In other words, was my choice of open subsets bad? Don't they form an atlas?
As discussed in the comments, you're on the right track and in order to prove that the stereographic projections are homeomorphisms, it's a good idea to simply write them (an their inverses) down explicitly.
As for your confusion about why they can be continuous even though 'the points on the left and on the right of the pole are mapped to very far away', recall what continuity actually means: It is enough that every point (say left of a pole) has a tiny neighbourhood which gets mapped closeby. You can always make the neighbourhood so small that it does not contain any point right from the pole. So everything is fine.