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

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 $0\leq|\alpha|\leq k$?

This question is motivated by the following ones:

• I think the answer is Yes, the proof is similar to the proof in Evans PDE 2nd page.269 (Extension theorem step 2).
– yoyo
Sep 21, 2016 at 14:04
• Evans proved the case for $k=1$, but the method are the same in general.
– yoyo
Sep 21, 2016 at 14:06
• @yoyo: I don't have any assumptions on $\partial\Omega$ for this question.
– user9464
Sep 21, 2016 at 22:56
• Oh!~~ You are right, the question is more complicated. Thanks for reminding.
– yoyo
Sep 22, 2016 at 11:05

Two different kinds of definitions of $C^k(\overline{\Omega})$ — extension and restriction.