Implicit Function Theorem

Theorem
Let $n$ and $m$ be natural numbers.

Let $\Omega \subset \R^n \times \R^k$ be open.

Let $f : \Omega \to \R^k$ be continuously differentiable.

Let $(a,b) \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.

Let $f(a,b) = 0$.

Let the $k\times k$ matrix
 * $\begin{pmatrix}\frac{\partial f}{\partial y^j}(a,b)\end{pmatrix}$

be nonsingular.

Then there exist niehgborhoods $U\subset\Omega$ of $a$ and $V\subset\R^k$ of $b$ such that there exists a unique continuously differentiable function $g : U \to V$ such that $f(x, g(x)) = 0$ for all $x\in U$.