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The classical inverse function theorem reads as follows:

Theorem 1 ($latex {C^1}&fg=000000$ inverse function theorem) Let $latex {\Omega \subset {\bf R}^n}&fg=000000$ be an open set, and let $latex {f: \Omega \rightarrow {\bf R}^n}&fg=000000$ be an continuously differentiable function, such that for every $latex {x_0 \in \Omega}&fg=000000$, the derivative map $latex {Df(x_0): {\bf R}^n \rightarrow {\bf R}^n}&fg=000000$ is invertible. Then $latex {f}&fg=000000$ is a local homeomorphism; thus, for every $latex {x_0 \in \Omega}&fg=000000$, there exists an open neighbourhood $latex {U}&fg=000000$ of $latex {x_0}&fg=000000$ and an open neighbourhood $latex {V}&fg=000000$ of $latex {f(x_0)}&fg=000000$ such that $latex {f}&fg=000000$ is a homeomorphism from $latex {U}&fg=000000$ to $latex {V}&fg=000000$.

It is also not difficult to show by inverting the Taylor expansion

$latex \displaystyle f(x) = f(x_0) + Df(x_0)(x-x_0) + o(\|x-x_0\|)&fg=000000$

that at each $latex {x_0}&fg=000000$, the local inverses $latex {f^{-1}: V \rightarrow U}&fg=000000$ are also differentiable at $latex {f(x_0)}&fg=000000$ with derivative

$latex \displaystyle Df^{-1}(f(x_0)) =…

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