# Implicit function theorem

Suppose $LaTeX: h:R^n \rightarrow R^m$, where $LaTeX: n > m$, and $LaTeX: h$ is in smooth. Further, suppose we can partition the variables, $LaTeX: x = (y, z)$, such that $LaTeX: y$ is m-dimensional with $LaTeX: \nabla_y h(x)$ nonsingular at $LaTeX: x^* = (y^*, z^*)$. Then, there exists $LaTeX: \varepsilon > 0$ for which there is an implicit function, $LaTeX: f$, on the neighborhood, $LaTeX: N_{\varepsilon}(z*) = \{z: \|z-z*\| < e\}$ such that $LaTeX: h(f(z), z)=0$ for all $LaTeX: z \in N_{\varepsilon}(z*)$. Further, $LaTeX: f$ is smooth with $LaTeX: \nabla_z f(z^*) = -\left( \nabla_y h(x^*) \right)^{-1} \nabla_z h(z^*)$.