This is cross-posted in MSE.

I have seen two different kinds of definitions of the notation $C^k(\overline{\Omega})$ — by "extension" of functions on $\Omega$ or by "restriction" of functions on $\mathbb{R}^n$. I'm not sure about how different these two kinds of definitions could be.

(I)Let the open set $\Omega\subset{\mathbb R}^n$, and $k$ be a positive integer. $C^k(\Omega)$ will denote the space of functions possessing continuous derivatives up to order $k$ on $\Omega$, and $C^k(\overline{\Omega})$ will denote the space of all $u\in C^k(\Omega)$ such that $\partial^{\alpha}u$ extends continuously to the closure $\overline{\Omega}$ for $0\leq|\alpha|\leq k$.

The above definition defines $C^k(\overline{\Omega})$ as a subset of $C^k(\Omega)$. See for instance Folland's *Introduction to Partial Differential Equations*.

On the other hand, one can also see in cited reference that $C^k(\overline{\Omega})$ is defined as restriction to $\overline{\Omega}$ of $C^k(\mathbb{R}^n)$ functions.

(II)For instance, in Hermann Sohr'sThe Navier-Stokes Equations — An Elementary Functional Analytic Approach(page.23), $C^k(\overline\Omega)$ means the space of all restrictions $u|_{\overline\Omega}$ to $\overline\Omega$ of functions $u\in C^k(\mathbb{R}^n)$ such that $$ \sup_{|\alpha|\leq k,x\in\mathbb{R}^n}|D^\alpha u(x)|<\infty. $$ Here $|\alpha|\leq k$ is replaced by $|\alpha|<\infty$ if $k=\infty$.

Here are my **questions**:

If $u\in C^k(\overline{\Omega})$ as in

(I), in general can $\partial^\alpha u$ extend continuously to $\mathbb{R}^n$ for $0\leq|\alpha|\leq k$? (*)[

Added thanks to Pietro Majer's comment below] If $u\in C^k(\overline{\Omega})$ as in(I), does there exist $w\in C^k(\mathbb{R}^n)$ such that $w|_\Omega=u$?- Are (I) and (II) essentially the same?