Continuity is fundamentally a topological property of a mapping. It just means that for a mapping F and a point p, for any neighborhood del of F(p), we can find a neighborhood eps of p such that F(eps) is contained entirely in del. In simpler terms, if you draw a little ball around F(p), I can find a little ball around p whose image under F is contained in the little ball you drew around F(p). If I have coordinates on the sphere that suddenly jump between 0 and 2*pi, I can’t satisfy this property, because points that are arbitrarily close on the sphere will be mapped to opposite sides of the “coordinate square” with sides [0,2*pi).

The Mercator projection is obtained by removing two points from the sphere (both poles) and stretching the hole at each pole until the punctured sphere forms a cylinder, then cutting the cylinder along a line of longitude. So you can see that the 3 discontinuities in the Mercator projection correspond to the top and bottom edges (where we poked a hole at each pole) and the left/right edges (where we cut the cylinder). (Note that stretching the sphere at the poles changes the curvature, but cutting the cylinder does not. The projection would have the same properties on a cylinder.)

It is possible to continuously map the sphere to the entire (infinite) plane if you just remove a single point (the north pole): place the sphere so the south pole is touching the origin of the plane and for any point on the sphere, draw a line from the north pole through that point. Where that line intersects the plane is that point’s image under this mapping (called the Riemann sphere).

That makes sense. As I thought, I just needed to understand continuity in this context. That also helps address my original question - why manifolds aren't widely used in cartography. Thank you.