Let $U,V \subset \mathbb{R}^n$ be open sets, and let $f: U \to V$ be a smooth bijection. Suppose that for every $x \in U$, the [Jacobian matrix](/page/Jacobian%20Matrix) $Jf_x \in \mathbb{R}^{n \times n}$ is invertible. Then $f$ is a Euclidean diffeomorphism; equivalently, the inverse map $f^{-1}: V \to U$ is smooth.