inverse function theorem

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

1. $𝐚\in X$, $𝐟(𝐚)\in Y$;

2. $Y=𝐟(X)$;

3. $𝐟:X\to Y$ is one-one;

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

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 $a\in I$, $f^{\prime }(a)\ne 0$, 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.