TITLE: Condition for $u\in C^k(\Omega)$ to be extendable to $u\in C^k(\overline{\Omega})$ for an open domain $\Omega\subset\Bbb R^n$
QUESTION [5 upvotes]: Fix some open $\Omega\subset\mathbb{R}^n$. Recall that for any subset $S\subset\mathbb{R}^n$, we define $$C^k(S)=\{u:S\to\mathbb{R}\ :\ \exists\hbox{ open }U\supseteq E,\ v\in C^k(U),\hbox{ s.t. }v\vert_S=u\}$$
Let $u\in C^k(\Omega)$ for some open $\Omega\subset\mathbb{R}^n$, such that for all $\lvert\alpha\rvert\le k$ we can extend $\partial^\alpha u$ continuously to $\overline{\Omega}$. My PDE professor claimed that this implies that $u\in C^k(\overline{\Omega})$, but I don't immediately see why this is true.
I succeeded in proving using Tietze's Extension Theorem that if $\partial_j u$ exists on $\Omega$ can be extended continuously to $\overline{\Omega}$, then $u$ can be extended to $\mathbb{R}^n$ such that $\partial_j u\in C(\mathbb{R}^n)$, but I have no way of controlling the different extensions given by different derivatives, it seems.

REPLY [3 votes]: It's true but it is far from obvious. You have to use Whitney's extension theorem, Whitney, which says that each function which satisfies a sort of Taylor formula of order $m$ on a closed set can be extended to a $C^m$ function in $\mathbb{R}^n$. See the link.