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.