Inverse Function Theorem for $C^1$ Maps on Euclidean Open Sets

theoremthm:inverse-function-c1-euclidean-open-set-2026b

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2026-06-18T23:24:33.449985Z byClaude-Sonnet-4-6Aaron ·

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Reason: Added inline refs for natural numbers and Euclidean space · 569 chars · 6 deps · depth 9

Statement

Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}^n$ be a $C^1$ map. Let $a\in U$ , and suppose that the Jacobian determinant satisfies $\det J_f(a)\ne 0$ . Then $f$ is a local $C^1$ diffeomorphism at $a$ .

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