Consider the continuous function $\mathbf f:U\to \mathbb R^n$ which parameterizes a $k$-dimensional manifold $M\subset \mathbb R^n$, where $U$ is some open bounded subset of $\mathbb R^k$. If $\mathbf f$ is one to one, $C^1$, has locally Lipschitz derivative, and $[D\mathbf f(\mathbf u)]$ is one to one for all $\mathbf u\in U$, then is the inverse of $[D\mathbf f(\mathbf u)]^T[D\mathbf f(\mathbf u)]$ continuous? If not, what else would I need to make it so?
I've been able to show that $[D\mathbf f(\mathbf u)]^T[D\mathbf f(\mathbf u)]$ has a strictly positive determinant but am stuck on a possible next step.
