Different Definitions on the Differentiability of Functions on a closed set. I have encountered three different definitions on the differentiablity of functions on a closed set.
In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The three definitions are


*

*We say $f: \bar\Omega\to \mathbb{R}$ belongs to $C^k(\bar\Omega)$, if there is a function $\tilde f\in C^k(\tilde\Omega)$ defined on a domain $\tilde \Omega \supset \Omega$ such that $f = \tilde f|_\Omega$.

*The function $f: \bar\Omega\to \mathbb{R}$ belongs to $C^k(\bar\Omega)$ if for each point $x\in \bar\Omega$, there is a neighborhood $U$ of $x$ and a $C^k$ function $\tilde f$ defined on $U$ such that $f|_U = \tilde f|_U$.

*Suppose that $\partial \Omega$ is the boundary of $\Omega$ and it is of $C^k$, that is for each point $x\in\partial\Omega$ there exists a neighborhood $U$ of $x$ and a $C^k$ diffeomorphism $\Phi: U \to V\subset\mathbb{R}^n$, where $V$ is a neighborhood of $0\in \mathbb{R}^n$, such that $\Phi(U\cap\Omega) = V\cap \{x\in\mathbb{R}^{n}|x^n >0\}$. Since for the part of the boundary $V\cap \{x\in\mathbb{R}^{n}|x^n =0\}$, the differentiablity is easy to defined using the directional derivative. Then we say $f\in C^k(\bar\Omega)$ is for each $x\in\omega$, $f\circ\Phi^{-1}$ is of $C^k$.


What's the relationship among the three definitions?
 A: I think all of  these are equivalent. Clearly (1) implies (2) because you can always choose $U = \tilde \Omega$. To show the converse, pick for every  $x \in \overline\Omega$ a neighborhood $U_x$ as in (2) and an extension $\tilde f_x \in C^k(U)$ of $f\vert_{\overline\Omega \cap U}$. Now set $\tilde\Omega = \bigcup_{x \in X} U_x$, pick a partition of unity $\{\phi_x\}_{x \in X}$ on $\tilde\Omega$ subordinate to the cover $\{U_x\}_{x \in X}$ and set $\tilde f = \sum_{x \in X} \phi_x \tilde f_x$. This function has the required properties.
In the case where $\partial \Omega$ is $C^k$ we also have the equivalence of (1) and (3). For this we need to show that the usual definition of a $C^k$ function on open subsets $U$ of the half-space $\mathbb{H}^n = \{x \in \mathbb{R}^n \mid x^n = 0\}$ agrees with definition (1). The implication (1) $\implies$ (2) is immediate. For the converse we can use a version of Borel's lemma$^1$. This guarantees the existence of a $C^k$ function $F: U \cap (\mathbb{R}^{n-1} \times \{0\}) \times (-1, 0]$ s.t. ${\partial_n}^kF(x, 0) = {\partial_n}^kf(x, 0)$ for all $x \in U \cap (\mathbb{R}^{n-1} \times \{0\})$. Hence the function
$$
\tilde f(x, t) = \begin{cases}
f(x, t) & t \geq 0 \\
F(x, t) & t \leq 0
\end{cases}
$$
defined on $U \cup \left(\left(U \cap \mathbb{R}^{n-1}\right) \times (-1, 0] \right)$ satisfies the required properties.
$^1$The  version proven in Wikipedia is not quite strong enough for this purpose. The version that we need is Corollary 1.3.4 in Hörmander's book The Analysis of Linear Partial Differential Operators I. The crucial difference is that Hörmanders version allows only requires $C^{k-j}$-regularity for the prescribed $j$-th derivative in $t$-direction.
