472: For Map Between Arbitrary Subsets of Euclidean C∞ Manifolds Ck at Point, Restriction on Domain That Contains Point Is Ck at Point

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A description/proof of that for map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, restriction on domain that contains point is $C^k$ at point

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Topics

About: $C^{\mathrm{\infty }}$ manifold

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof

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Starting Context

• The reader knows a definition of map between arbitrary subsets of Euclidean $C^{\mathrm{\infty }}$ manifolds $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.

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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 $C^{\mathrm{\infty }}$ manifolds $C^k$ at any point, where $k$ includes $\mathrm{\infty }$, the restriction on any domain that contains the point is $C^k$ at the point.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

For any Euclidean $C^{\mathrm{\infty }}$ manifolds, ${\mathbb{R}}^{d_1},{\mathbb{R}}^{d_2}$, any subsets, $S_1,S_1^{\prime }\subseteq {\mathbb{R}}^{d_1},S_2\subseteq {\mathbb{R}}^{d_2}$, such that $S_1^{\prime }\subseteq S_1$, any point, $p\in S_1^{\prime }$, any natural number (including 0) or $\mathrm{\infty }$ $k$, and any map, $f:S_1\to S_2$, such that $f$ is $C^k$ at $p$, $f{|}_{S_1^{\prime }}:S_1^{\prime }\to S_2$ is $C^k$ at $p$.

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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 $f(U_p)\subseteq U_{f(p)}$. $U_p\cap S_1^{\prime }\subseteq S_1^{\prime }$ is an open neighborhood of $p$ on $S_1^{\prime }$, and $f{|}_{S_1^{\prime }}(U_p\cap S_1^{\prime })\subseteq U_{f(p)}$.

Let us suppose that $1\le k$ including $\mathrm{\infty }$.

There are an open neighborhood, $U_p^{\prime }\subseteq {\mathbb{R}}^{d_1}$, of $p$ and a map, $f^{\prime }:U_p^{\prime }\to {\mathbb{R}}^{d_2}$, $C^k$ at $p$ such that $f^{\prime }{|}_{U_p^{\prime }\cap S_1}=f{|}_{U_p^{\prime }\cap S_1}$. $U_p^{\prime }$ and $f^{\prime }$ can be used also for $f{|}_{S_1^{\prime }}$, because $f^{\prime }{|}_{U_p^{\prime }\cap S_1^{\prime }}=f{|}_{S_1^{\prime }}{|}_{U_p^{\prime }\cap S_1^{\prime }}$.

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References

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