The following is a proof from the book Introduction to manifolds by Tu ( Page 63)
I did not understand the highlighted step. Please explain.
2025-01-13 02:08:26.1736734106
How did the smoothness of map into R^m transfer to the smoothness of map into R
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Notice: $$Fo\phi ^{-1}:\mathbf R^n \to \mathbf R^m$$ More precisely domain is subset of $\mathbf R^n$ and n is dimension of manifold, so when this map that you've see in multivariable calculus is $C^{\infty}$ in a usual definition? When all his component functions that are $\mathbf R^n \to \mathbf R$ are $C^{\infty}$.