Implicit Function Theorem

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The Implicit Function Theorem gives conditions on derivatives which ensure that an implicitly defined set is the graph of a function.

Real Functions

Let $n$ and $k$ be natural numbers.

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

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

Let the partial derivatives of $f$ with respect to $\R^k$ be continuous.

Let $\tuple {a, b} \in \Omega$, with $a\in \R^n$ and $b\in \R^k$.

Let $\map f {a, b} = 0$.

For $\tuple {x_0, y_0} \in \Omega$, let $D_2 \map f {x_0, y_0}$ denote the total derivative of the function $y \mapsto \map f {x_0, y}$ at $y_0$.

Let the linear map $D_2 \map f {a, b}$ 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 $\map f {x, \map g x} = 0$ for all $x \in U$.

Moreover, $g$ is continuous.


Unit Circle

Consider the Equation of Unit Circle for the unit circle:

$x^2 + y^2 = 1$

This is the graph of a function near all points where $y = 0$.

Also see