Can $C^k(\overline{\Omega})$ functions extend to $C^k(\mathbb{R}^n)$?

Question

Consider the following definition.

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 $0\leq|\alpha|\leq k$.

Here is my question:

If $u\in C^k(\overline{\Omega})$, can $\partial^\alpha u$ extend continuously to $\mathbb{R}^n$ for $0\leq|\alpha|\leq k$?

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This question is motivated by the following ones:

• Relationship among the function spaces $C_c^\infty(\Omega)$, $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Bbb{R}^d)$
• What does the notation $C(\bar U)$ mean for $U\subset\Bbb{R}^d$ open?
• Reference request: $C^k(\overline\Omega)$ as restriction of $C^{k}(\mathbb{R}^d)$ functions on $\Omega$

Answer 1

This post has been put as a more complete one in MO:
Two different kinds of definitions of $C^k(\overline{\Omega})$ — extension and restriction.

Thanks to a comment there, the answer to the question is yes. But the question in this post does not match the one in the title. See more details in the accepted MO answer.