Proving a problem by identity theorem

Question

In the proof of this theorem it said that f1 is non-constant(by identity theorem). I tried to understand why this is true and I wrote the solution and obtained that restriction of f to U1 is constant,now how can I prove U1 has limit point and then by identity theorem the proof will be complete because it is a contradiction?

Answer 1

By definition of a chart, the sets $U$ and $V$ are open. All points inside are limit points.