Inverse Function Theorem/Examples/Square Function

Example of Use of Inverse Function Theorem

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