Definition of the space of Hölder continuous functions I have a question concerning the definition of the space of Hölder continuous functions $C^{k,\alpha}(\Omega)$, where $\Omega\subset\mathbb{R}^n$ is an open set and $k\in\mathbb{N}$ and $\alpha\in (0,1]$.
Often the following definition is used: $f\in C^{k,\alpha}(\Omega)$ if and only if $f\in C^{k}(\Omega)$ such that all its derivatives up to order $k$ are bounded functions and the $k-th$ order derivatives are Hölder continuous.
I am wondering why only $D^{\beta}f:U\rightarrow \mathbb{R}$ with $\vert \beta \vert=k$ has to be Hölder continuous? Does it maybe follow that $f$ and its derivatives up to order $k-1$ have to be Hölder continuous as well? Otherwise the space of Hölder continuous functions would contain functions which are not Hölder continuous, which sounds very awkward to me.
Best wishes
 A: It is possible to further weaken the assumptions given on $\Omega$ as being "only" a bounded domain with $C^1$ boundary (even Lipschitz should still work).
It suffices to prove that if $u \in C^{1,\alpha}_b(\Omega)$, i.e. $u$, and $\partial_i u$ are bounded and $\partial_i u$ are $\alpha$-Hölder, then $u$ is $\alpha$-Hölder as well.
Since $\Omega$ is a bounded $C^1$-domain, there exist finitely many bounded diffeomorphisms $\Phi_i \colon \mathbb R^n \to \mathbb R^n$ such that $$\Phi_i(\Omega \cap U_i) \subset B_i^+, \quad \Phi(\partial\Omega \cap U_i) = B_i^+ \cap (\mathbb R^{n-1} \times \{0\})$$
Where $(U_i)_{i = 1,\dots,N}$ is an open covering of $\partial\Omega$ and 
$$
B_i^+ := \{ x  \in \mathbb R^n \colon |x| < r_i, x_n \geq 0\}.
$$
Correct me if something seems odd here, as this should be standard for $C^1$-submanifolds.
Now, take $x,y \in \Omega$ and let $\delta > 0$ be the Lebesgue number of the covering $(U_i)$. Now we distinguish the following cases


*

*If $|x - y| < \delta$ and $x,y \in \overline{B_\delta(x)} \subset \Omega$: In this case, the mean value theorem is applicable and yields
$$
|u(x) - u(y)| \leq \|\nabla u\|_\infty |x - y|.
$$

*If $|x - y| < \delta$ and $x,y \in \Omega \cap U_i$: Consider the convex hull
$$
W := \{(1 - t) \Phi_i(x) + t \Phi(y) \colon t \in [0,1]\} \subset B_i^+.
$$
Then
\begin{align*}
|u(x) - u(y)| 
&= |u(\Phi_i^{-1}\circ \Phi_i(x)) - u(\Phi_i^{-1}\Phi_i(y))| \\
&= \Big|\int_0^1 \frac{d}{dt} \Big((u\circ\Phi_i^{-1})\big((1 - t)\Phi_i(x) + t \Phi_i(y)\big) \Big) d t \Big| \\
&\leq \| \nabla(u \circ \Phi_i^{-1})\|_\infty \cdot |\Phi_i(x) - \Phi_i(y)| \\
&\leq \| \nabla u\|_{\infty} \cdot \| D\Phi_i^{-1}\|_\infty \cdot \|D\Phi_i\|_{\infty} |x - y|,
\end{align*}
where we applied the mean value theorem to $\Phi_i$.

*If $|x - y| \geq \delta$, then
$$
|u(x) - u(y)| \leq 2 \|u\|_\infty \cdot 1 \leq 2 \|u\|_{\infty} \frac{|x - y|}{\delta}.
$$
Hence, 
$$
\sup_{x, y \in \Omega, x \neq y} \frac{|u(x) - u(y)|}{|x - y|} \leq C (\|u\|_\infty + \|\nabla u\|_\infty)
$$
for some constant $C > 0$ and thus $u$ is Lipschitz, i.e. in particular $\alpha$-Hölder.
