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Suppose that $f:O\to \mathbb R^n$ is a smooth change of variables on the open subset O of $\mathbb R^n$. Let D be an open subset of $\mathbb R^n$ such that $D\cup \partial D$ is contained in O. Show that $f(\partial D)$ = $\partial f(D)$.

My attempt was to try use the Inverse Function Theorem, but I don't know how to start. A smooth change of variable is that $f$ is one to one and its Jacobian is not zero.

Sorry for my english.

João Alves Jr.
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Rodrigo Palacios
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  • What do you mean by "a smooth change of variables"? Does the map $f$ have to be a diffeomorphism? Does it have to be regular (i.e. Jacobian nowhere zero)? – Ernie060 Oct 28 '18 at 17:27
  • You mean Jacobian non-zero, I guess? Thanks for the clarification. – Ernie060 Oct 28 '18 at 20:37
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    By the inverse function theorem, $f$ has a differentiable local inverse around every point of its range. Since $f$ is one-to-one, these local inverses paste together to a global inverse $range(f)\to O$, so $f$ is a diffeomorphism, consequently a homeomorphism, and as such, it preserves the topological structures. – Berci Oct 28 '18 at 21:41

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