Need periodic review!

The definition of manifold itself implies the continuity of each coordinate map, which is due to the differentiability or smoothness property of the overlap map $f_{UV}$ and $f_{VU}$. Smoothness is a much stronger property then continuity. To be added, the coordinate map, the injective map from an open subset of $M^n$ into $\mathbb{R}^n$, i.e. a Euclidean space, is in fact equivalent to the realization of a quantum state to a particular representation.

In addition, it can be seen that due to the unfamiliarity with the mathematical definitions and theorems, a naive and obvious fact was misunderstood by me as a big and serious problem. How funny and silly!

It is therefore both mandatory and urgent to perform a thorough, repetitive and periodic review!