Implicit Function Theorem for Differentiable Real Functions

Theorem

Let $\Omega \subset \R^{n+k}$ be open.

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

Let the $i$th partial derivatives of $f$ be continuous in $\Omega$ for $n+1 \leq i \leq n+k$.

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

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

For $(x_0,y_0)\in\Omega$, let $D_2 f(x_0,y_0)$ denote the differential of the function $y\mapsto f(x_0, y)$ at $y_0$.

Let the linear map $D_2 f(a,b)$ be invertible.

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

Moreover, $g$ is differentiable, and its differential satisfies:

$dg (x) = - \left( (D_2f)(x, g(x)) \right)^{-1} \circ (D_1 f)(x, g(x))$ for all $x\in U$.

Proof

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