Inverse function theorem in R^k

Inverse function theorem in $\mathbb R^k$: Let $U\subseteq\mathbb R^k$ be an open set and let $f:U\to\mathbb R^k$ be continuously differentiable. If the Jacobian determinant $\det Df(a)$ is nonzero at some $a\in U$, then there exist neighborhoods $A$ of $a$ and $B$ of $f(a)$ such that $f$ restricts to a bijection $f:A\to B$ whose inverse $f^{-1}:B\to A$ is continuously differentiable. Moreover,

Thus, near $a$, the map $f$ is a diffeomorphism onto its image; in particular it is a local homeomorphism .