# Boundary of the image is the image of the boundary.

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

Sorry for my english.

• 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
• 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