Implicit function theorem

Let $\mathcal{O}\subset {\mathbb{R}}^{n+k}$ be open with $F:\mathcal{O}\to {\mathbb{R}}^n$ continuously differentiable. Also let $(x_0,y_0)\in \mathcal{O}$ such that $F(x_0,y_0)=0$ and the $k\times k$ partial derivative matrix $D_{y}F(x_0,y_0)$ is invertible.

Then,

1. there exists $r>0$ and an open ball $B=B_r(x_0)$ with a continuously differentiable map $G:B\to {\mathbb{R}}^k$ such that

$F(x,G(x))=0\forall x\in B$

2. when $||x-x_0||<r$, $||y-y_0||<r$, and $F(x,y)=0$ then

$y=G(x)$ 3.
$D_{x}F(x,G(x))+D_{y}F(x,G(x))\cdot DG(x)=0\forall x\in B$