Inverse Function Theorem

Theorem

Let $n \in \N$ be a natural number.

Let $f: \R^n \to \R^n$ be a mapping on the real Cartesian space of $n$ dimensions.

Let $\mathbf x \in \R^n$ be an element of $\R^n$.

Let the Jacobian matrix of $f$ be non-singular in the locality of $\mathbf x$.

Then there exists a local single-valued differentiable inverse for $f$ at the locality of $\mathbf x$.

Proof

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Examples

Square Function

The real function $f: \R \to \R$ defined as:

$\forall x \in \R: \map f x = x^2$

does not have a local differentiable inverse around $x = 0$, because $\map f 0 = 0$.

However, it does have a local differentiable inverse around every $a \ne 0$, because $\map f a \ne 0$.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): inverse function theorem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): inverse function theorem