A description/proof of that for map between arbitrary subsets of Euclidean \(C^\infty\) manifolds \(C^k\) at point, restriction on domain that contains point is \(C^k\) at point
Topics
About: \(C^\infty\) manifold
The table of contents of this article
Starting Context
Target Context
- The reader will have a description and a proof of the proposition that for any map between any arbitrary subsets of any Euclidean \(C^\infty\) manifolds \(C^k\) at any point, where \(k\) includes \(\infty\), the restriction on any domain that contains the point is \(C^k\) at the point.
Orientation
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Main Body
1: Description
For any Euclidean \(C^\infty\) manifolds, \(\mathbb{R}^{d_1}, \mathbb{R}^{d_2}\), any subsets, \(S_1, S'_1 \subseteq \mathbb{R}^{d_1}, S_2 \subseteq \mathbb{R}^{d_2}\), such that \(S'_1 \subseteq S_1\), any point, \(p \in S'_1\), any natural number (including 0) or \(\infty\) \(k\), and any map, \(f: S_1 \to S_2\), such that \(f\) is \(C^k\) at \(p\), \(f \vert_{S'_1}: S'_1 \to S_2\) is \(C^k\) at \(p\).
2: Proof
Let us suppose that \(k = 0\).
For any open neighborhood, \(U_{f (p)} \subseteq S_2\), of \(f (p)\), there is an open neighborhood, \(U_p \subseteq S_1\), of \(p\) such that \(f (U_p) \subseteq U_{f (p)}\). \(U_p \cap S'_1 \subseteq S'_1\) is an open neighborhood of \(p\) on \(S'_1\), and \(f \vert_{S'_1} (U_p \cap S'_1) \subseteq U_{f (p)}\).
Let us suppose that \(1 \le k\) including \(\infty\).
There are an open neighborhood, \(U'_p \subseteq \mathbb{R}^{d_1}\), of \(p\) and a map, \(f': U'_p \to \mathbb{R}^{d_2}\), \(C^k\) at \(p\) such that \(f' \vert_{U'_p \cap S_1} = f \vert_{U'_p \cap S_1}\). \(U'_p\) and \(f'\) can be used also for \(f \vert_{S'_1}\), because \(f' \vert_{U'_p \cap S'_1} = f \vert_{S'_1} \vert_{U'_p \cap S'_1}\).
