Prove differential homeomorphism, and prove homeomorphism first. This example is obvious. Then it is proved that the mapping is differentiable. You made a mistake here, that is, you confused the original space with the coordinate card. As a topological space, the original manifold is nondifferentiable (even R in the example is just a straight line when you regard it as a topological space, so you should forget its differential structure). Only if you are locally homeomorphic in Euclidean space, will you borrow the differential structure in Euclidean space as your local differential structure (differential structure is an additional structure). Therefore, to examine whether two differential manifolds are differential homeomorphisms, it is necessary to examine the mapping between two coordinate cards, not between two topological spaces (the mapping of the topological space given in the example is not differentiable in principle, and you think it is differentiable because you have not forgotten its original differential structure). That is, to investigate whether the mapping obtained by combining the upper coordinate mapping before and after the topological space mapping is differentiable. In this example, the composite map is actually an identity map, so it is naturally smooth.
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