I'm reading a book and I discovered a strange (to me) statement:
(roughly stated): $\Omega$ is bounded open set in $\mathbb{R}^d$. Then
(i) If $u \in C^{0,a}(\Omega)$, then "statement 1" holds.
(ii) If $u \in C^{0,a}(\overline{\Omega})$, then "statement 2" holds. \qed
As far as I know, if $u$ belongs to $C^{0,a}(\Omega)$, then it is uniformly continuous on $\Omega$. Then, $u$ can be continuously extended to $\partial \Omega$, and hence $u \in C^{0,a}(\overline{\Omega})$.
Am I right? I'm very confused as to why the author wrote such distinguished statements as (i) and (ii). Is there an essential difference between $C^{0,a}(\Omega)$ and $C^{0,a}(\overline{\Omega})$? (when $\Omega$ is open and bounded).
If $u\in C^{0,\alpha}(\Omega)$, then you can $\textbf{continuously}$ extend it to $\overline{\Omega}$, which only means that $u\in C(\overline{\Omega})$, not that $u\in C^{0,\alpha}(\overline{\Omega})$, as the latter requires that $$ \sup_{x,y\in\overline{\Omega}, x\neq y} \frac{\vert u(x) - u(y) \vert}{\vert x - y \vert^\alpha} < \infty.$$