inverse function theoremInverseFunctionTheorem2002-08-24 05:47:452002-12-28 15:18:18Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $\mathbf{f}$ 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
\begin{enumerate}
\item $\mathbf{a} \in X$, $\mathbf{f}(\mathbf{a}) \in Y$;
\item $Y = \mathbf{f}(X)$;
\item $\mathbf{f}:X \to Y$ is one-one;
\item $\mathbf{g}$ is continuously differentiable on $Y$ and $\mathbf{g}(\mathbf{f}(\mathbf{x})) = \mathbf{x}$ for all $\mathbf{x} \in X$.
\end{enumerate}
\subsubsection{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 $a \in I$, $f'(a) \neq 0$, 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 $[f(\alpha), f(\beta)]$.
\subsubsection{Note} The inverse function theorem is a special case of the implicit function theorem where the dimension of each variable is the same.