467: Map from Open Subset of Euclidean C∞ Manifold into Subset of Euclidean C∞ Manifold Ck at Point, Where k Excludes 0 and Includes ∞

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definition of map from open subset of Euclidean $C^{\mathrm{\infty }}$ manifold into subset of Euclidean $C^{\mathrm{\infty }}$ manifold $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$

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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: Structured Description
• 2: Natural Language Description
• 3: Note

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Starting Context

• The reader knows a definition of Euclidean $C^{\mathrm{\infty }}$ manifold.
• The reader knows a definition of map.

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Target Context

• The reader will have a definition of map from open subset of Euclidean $C^{\mathrm{\infty }}$ manifold into subset of Euclidean $C^{\mathrm{\infty }}$ manifold $C^k$ at point, where $k$ excludes $0$ and includes $\mathrm{\infty }$.

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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: Structured Description

Here is the rules of Structured Description.

Entities:
${\mathbb{R}}^{d_1}$: $=\text{ the Euclidean }{C}^{\mathrm{\infty }}\text{ manifold }$
${\mathbb{R}}^{d_2}$: $=\text{ the Euclidean }{C}^{\mathrm{\infty }}\text{ manifold }$
$U$: $\in \{\text{ the open subsets of }{\mathbb{R}}^{d_1}\}$
$S$: $\in \{\text{ the subsets of }{\mathbb{R}}^{d_2}\}$
$p$: $\in U$
$k$: $\in (\mathbb{N}\setminus \{0\})\cup \{\mathrm{\infty }\}$
$\ast f$: $:U\to S$
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Conditions:
$\mathrm{\exists }{U}_p\in \{\text{ the open neighborhoods of }p\text{ on }{\mathbb{R}}^{d_1}\}\text{ such that }{U}_p\subseteq U(f\text{ 's all the up-to-}k\text{-th partial derivatives (when }k=\mathrm{\infty }\text{, that means all the partial derivatives of all the orders) exist on }{U}_p\wedge \text{ the derivative maps as the maps from }{U}_p\text{ into }{\mathbb{R}}^{d_2}\text{are continuous on }{U}_p)$
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2: Natural Language Description

For any Euclidean $C^{\mathrm{\infty }}$ manifolds, ${\mathbb{R}}^{d_1},{\mathbb{R}}^{d_2}$, any open subset, $U\subseteq {\mathbb{R}}^{d_1}$, any subset, $S\subseteq {\mathbb{R}}^{d_2}$, any point, $p\in U$, and any natural number (excluding 0) or $\mathrm{\infty }$ $k$, any map, $f:U\to S$, such that there is an open neighborhood, $U_p\subseteq {\mathbb{R}}^{d_1}$, of $p$ such that $U_p\subseteq U$ and $f$'s all the up-to-$k$-th partial derivatives (when $k=\mathrm{\infty }$, that means all the partial derivatives of all the orders) exist on $U_p$ and the derivative maps as the maps from $U_p$ into ${\mathbb{R}}^{d_2}$ are continuous on $U_p$

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3: Note

$k=0$ is excluded because that case has been already defined as map continuous at point.

But when $f$ is $C^k$ at $p$ where $1\le k$, $f$ is $C^0$ at $p$: for any open neighborhood, $U_{f(p)}\subseteq S$, of $f(p)$, $U_{f(p)}=U_{f(p)}^{\prime }\cap S$ where $U_{f(p)}^{\prime }\subseteq {\mathbb{R}}^{d_2}$ is open, and as $f{|}_{U_p}:U_p\to {\mathbb{R}}^{d_2}$ is $C^k$, $f{|}_{U_p}$ is continuous at $p$, and there is an open neighborhood, $U_p^{\prime }\subseteq U_p\subseteq U$, such that $f{|}_{U_p}(U_p^{\prime })\subseteq U_{f(p)}^{\prime }$, but $f(U_p^{\prime })\subseteq U_{f(p)}^{\prime }\cap S=U_{f(p)}$.

$f$ is required to be from an open subset of ${\mathbb{R}}^n$ in order for the derivatives of $f$ to exist at $p$.

There is a definition for the case that the domain is an arbitrary subset of ${\mathbb{R}}^n$.

Being called "$C^k$ at $p$", the derivatives are required to exist on the whole $U_p$, because in addition to that the up-to-$k-1$-th partial derivatives have to exist on a neighborhood of $p$ in order for the $k$-th partial derivatives to exist, judging the continuousness of the derivative maps as the maps from $\{p\}$ is meaningless as being vacuously continuous as the constant maps.

There can be a weaker definition that requires the continuousnesses only at $p$, which we have not adopted, because we do not particularly need that "continuous only at $p$" case.

$S$ does not need to be open on ${\mathbb{R}}^{d_2}$, because taking the derivatives or judging the continuousnesses does not require $S$ to be open.

For the $k=\mathrm{\infty }$ case, the definition requires that a common $U_p$ exists for all the $k$-th partial derivatives, while there can be a weaker definition that requires that a $U_{p,k}$ exists for each $k$, which is at least directly different from our definition, because ${\cap }_k{U}_{p,k}$ may not be open unless proved otherwise.

When $f$ is $C^k$ at each $p\in U$, $U_p$ can be taken to be $U$: while $U={\cup }_{p\in U}{U}_p$, for each $p\in U_{p^{\prime }}\cap U_{p^{″}}$, each $j$-th partial derivative at $p$ on $U_{p^{\prime }}$ equals that on $U_{p^{″}}$, and so, there is the derivative map from $U$ into ${\mathbb{R}}^{d_2}$ as the compound of the derivative maps from $U_p$ s (means that each domain restriction of the derivative map from $U$ is the derivative map from $U_p$), which (the derivative map from $U$) is continuous, by the proposition that any map between topological spaces is continuous if the domain restriction of the map to each open set of a possibly uncountable open cover is continuous.

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References

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