= Solution
The <inverse function theorem> states: let $F:U\subseteq\mathbb R^n\to\mathbb R^n$ be $C^1$, with $U$ open, and suppose $DF(x_0)$ is invertible. Then there exist open neighbourhoods $U_0\subseteq U$ of $x_0$ and $V_0$ of $F(x_0)$ such that $F:U_0\to V_0$ is a $C^1$ <diffeomorphism>. Its inverse derivative satisfies
$$
\boxed{D(F^{-1})(F(x))=(DF(x))^{-1},\qquad x\in U_0.}
$$
The conclusion is local; invertibility of one <Jacobian matrix> does not assert global injectivity.
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