Inverse Function Theorem

Definition

For a continuously differentiable function \(\boldsymbol{\mathcal{f}}\), if the Jacobian matrix \(\mathbf{J_{\mathcal{f}}}\) defined by \(\{ \mathbf{J_{\mathcal{f}}}(\mathbf{x}) \}_{ij} = \partial f_{i}(\mathbf{x}) / \partial x_{j}\) is nonsingular at a point \(\mathbf{x}^{*}\), then there is a neighbourhood of \(\boldsymbol{\mathcal{f}}(\mathbf{x}^{*})\) in which \(\boldsymbol{\mathcal{f}}\) is invertible.