Problems. 2-36.* Let $A \subset \mathbb{R}^n$ be an open set and $f: A \to \mathbb{R}^n$ a continuously differentiable 1-1 function such that $det f'(x) \neq 0$ for all $x$. Show that $f(A)$ is an open set and $f^{-1}: f(A) \to A$ is differentiable. Show also that $f(B)$ is open for any open set $B \subset A$.
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Let A ⊂ R^n be open and f: A → R^n be C^1 and 1-1 with det f'(x) ≠ 0 for every x ∈ A. Use the Inverse Function Theorem at each point of A to obtain local diffeomorphisms; then patch these local results to get the conclusions. Show more…
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