# inverse function theorem

Let $\mathbf{f}$ be a continuously differentiable, vector-valued function mapping the open set $E\subset\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ and let $S=\mathbf{f}(E)$. If, for some point $\mathbf{a}\in E$, the Jacobian, $|J_{\mathbf{f}}(\mathbf{a})|$, is non-zero, then there is a uniquely defined function $\mathbf{g}$ and two open sets $X\subset E$ and $Y\subset S$ such that

1. 1.

$\mathbf{a}\in X$, $\mathbf{f}(\mathbf{a})\in Y$;

2. 2.

$Y=\mathbf{f}(X)$;

3. 3.

$\mathbf{f}:X\to Y$ is one-one;

4. 4.

$\mathbf{g}$ is continuously differentiable on $Y$ and $\mathbf{g}(\mathbf{f}(\mathbf{x}))=\mathbf{x}$ for all $\mathbf{x}\in X$.

## 0.0.1 Simplest case

When $n=1$, this theorem becomes: Let $f$ be a continuously differentiable, real-valued function defined on the open interval $I$. If for some point $a\in I$, $f^{\prime}(a)\neq 0$, then there is a neighbourhood $[\alpha,\beta]$ of $a$ in which $f$ is strictly monotonic. Then $y\to f^{-1}(y)$ is a continuously differentiable, strictly monotonic function from $[f(\alpha),f(\beta)]$ to $[\alpha,\beta]$. If $f$ is increasing (or decreasing) on $[\alpha,\beta]$, then so is $f^{-1}$ on $[f(\alpha),f(\beta)]$.

## 0.0.2 Note

The inverse function theorem is a special case of the implicit function theorem where the dimension of each variable is the same.

Title inverse function theorem InverseFunctionTheorem 2013-03-22 12:58:30 2013-03-22 12:58:30 azdbacks4234 (14155) azdbacks4234 (14155) 9 azdbacks4234 (14155) Theorem msc 03E20 DerivativeOfInverseFunction LegendreTransform DerivativeAsParameterForSolvingDifferentialEquations TheoryForSeparationOfVariables