The Implicit Function Theorem

In the setting of Functions of Class Cᵏ: Notation, with points $(x,y)$ of $\mathbb{R}^{n+m}$, sets $A\times B$ and blocks $D_{x}F$, $D_{y}F$ of the Jacobian matrix as in Functions of Class Cᵏ: Notation (vi), let $k\in\mathbb{N}$, let $U$ be an open subset of $\mathbb{R}^{n+m}$ and let $F:U\to\mathbb{R}^{m}$ be of class $C^{k}$ on $U$. Let $(a,b)\in U$ satisfy $F(a,b)=0$, with $D_{y}F(a,b)$ invertible. Then there are open sets $A\subseteq\mathbb{R}^{n}$ and $B\subseteq\mathbb{R}^{m}$ with $a\in A$, $b\in B$ and $A\times B\subseteq U$, and a function $g:A\to B$, with the following properties.

(i) Implicit function. For all $x\in A$ and $y\in B$,

$$F(x,y)=0\quad\Longleftrightarrow\quad y=g(x).$$

In particular $g(a)=b$ and $F(x,g(x))=0$ for every $x\in A$.

(ii) Uniqueness. $g$ is the only function $h:A\to B$ with $F(x,h(x))=0$ for every $x\in A$.

(iii) Regularity. $g$, regarded as a function $A\to\mathbb{R}^{m}$, is of class $C^{k}$ on $A$, the matrix $D_{y}F(x,g(x))$ is invertible for every $x\in A$, and

$$Dg(x)=-D_{y}F\bigl(x,g(x)\bigr)^{-1}D_{x}F\bigl(x,g(x)\bigr)\qquad\text{for every }x\in A .$$