implicit function theorem

Theorem.

Let $\mathrm{\Omega }$ be an open subset of ${\mathrm{R}}^n\mathrm{\times }{\mathrm{R}}^m$and let $f\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{(}\mathrm{\Omega }\mathrm{,}{\mathrm{R}}^m\mathrm{)}$. Let $\mathrm{(}{x}_{\mathrm{0}}\mathrm{,}{y}_{\mathrm{0}}\mathrm{)}\mathrm{\in }\mathrm{\Omega }\mathrm{\subset }{\mathrm{R}}^n\mathrm{\times }{\mathrm{R}}^m$.If the matrix $D_y\mathit{}f\mathit{}\mathrm{(}{x}_{\mathrm{0}}\mathrm{,}{y}_{\mathrm{0}}\mathrm{)}$ defined by

$$D_{y}f(x_0,y_0)={\left(\frac{\partial f_j}{\partial y_k}(x_0,y_0)\right)}_{j,k}\mathit{\hspace{1em}}j=1,\mathrm{\dots },m\mathit{\hspace{1em}}k=1,\mathrm{\dots },m$$

is invertible, then there exists a neighborhood $U\mathrm{\subset }{\mathrm{R}}^n$ of $x_{\mathrm{0}}$and a function $g\mathrm{\in }{C}^{\mathrm{1}}\mathit{}\mathrm{(}U\mathrm{,}{\mathrm{R}}^m\mathrm{)}$ such that

$$f(x,g(x))=f(x_0,y_0)\mathit{\hspace{1em}\hspace{1em}}\forall x\in U.$$

Moreover

$$Dg(x)=-{(D_{y}f(x,g(x)))}^{-1}{D}_{x}f(x,g(x)).$$