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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

  1. 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 $f = \tilde f|_\Omega$.
  2. 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 $f|_U = \tilde f|_U$.
  3. 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?

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