> [!definition] > > Let $X$ and $Y$ be $C^p$-[[Manifold|manifolds]], and $A \subset X$. A mapping $f: A \to Y$ is **$C^p$ on** $A$, if for any $p \in A$, there exists an [[Open Set|open]] [[Neighbourhood|neighbourhood]] $U \in \cn^o(p)$ on which $f$ has a $C^p$-extension. > [!theorem] > > Let $X$ be a [[n-Manifold|n-manifold]] of class $C^p$, $A \subset X$ be [[Closed Set|closed]], and $f: A \to \real^k$ be $C^p$ on $A$, then for any $U \supset A$ [[Open Set|open]], there exists a $C^p$ extension $F: X \to \real^k$ such that $F|_{A} = f$ and $\supp{F} \subset U$. > > *Proof*. Let $p \in A$, then there exists $U_p \in \cn^o(p)$ and $f_p: U_p \to \real^k$ smooth such that $f_p|_A = f$. Assume without loss of generality that $U_p \subset U$, then the family $\bracs{U_p: p \in A}$ combined with $A^c$ forms an [[Open Cover|open cover]] of $X$. Let $\bracs{\psi_p: p \in A}$ and $\psi_0$ be a [[Partition of Unity on n-Manifold|smooth partition of unity]] subordinate to it with $\supp \psi_p \subset U_p \subset U$ for each $p$, and $\supp{\psi_0} \subset A^c$. Define > $ > F = \sum_{p \in A}\psi_p \cdot f_p > $ > Let $p \in X$, then there exists $V \in \cn^o(p)$ such that the sum is non-zero on only finitely many terms. Since each term is smooth, $F$ is smooth at $p$. > > Moreover, for any $x \in A$, > $ > F(x) = \sum_{p \in A}\psi_p(x) \cdot f_p(x) = \sum_{p \in A}\psi_p(x) \cdot f(x) = f(x) > $ > because the non-$\psi_0$ functions sum up to $1$, so $F|_A = f$. > > Lastly, as $\supp{\psi_p} \subset U_p \subset U$ for each $p$, $\supp{F} \subset U$.