What functions are in the set $$ c^\alpha \setminus \bigcup_{\beta \in (\alpha,1)} C^\beta?$$
A single example will do, but the more the merrier.
This question was natural to me after writing this answer; all I did was summarise info that was on MO. There, it is explained that $x^\alpha$ on $[0,1]$ belongs to the set $C^\alpha \setminus c^\alpha$.
What is $c^\alpha$? It is the 'little $\alpha$-Hölder space', either defined as the closure of $C^\infty$ in the $C^\alpha$ norm, or equivalently as the subset of $C^\alpha$ functions for which $\lim_{t\downarrow 0} \sup_{0<|x-y|<t} \frac{|f(x) - f(y)|}{|x-y|^\alpha} = 0.$ Naturally there are the inclusions
$$ C^\infty \subset c^\alpha \subset C^\alpha$$
The question of what functions are in $c^\alpha \setminus C^\infty$ is trivial. If $\alpha < \beta$ then we can further write
$$ C^\infty \subset c^\beta \subset C^\beta \subset c^\alpha \subset C^\alpha$$
For fixed $0<\alpha<\beta<1$, it is also trivial to see that $ x^{(\alpha+\beta)/2}$ is in $c^\alpha \setminus C^\beta([0,1])$. The functions $x^\alpha$ are also examples of functions in $C^\alpha \setminus \bigcup_{\beta \in (\alpha,1)} C^\beta$ (this is already implied by being in $C^\alpha \setminus c^\alpha$).
Approaching from the other direction is also not interesting: the function $(x \log x)^\alpha$ on $[0,1/2]$ is easily seen to be a function that belongs to the set $$\bigcap_{\beta \in (0, \alpha) } C^\beta \setminus C^\alpha = \bigcap_{\beta \in (0, \alpha) } c^\beta \setminus C^\alpha \subset \bigcap_{\beta \in (0, \alpha) } c^\beta \setminus c^\alpha.$$
I think therefore, this question is the simplest non-trivial question that can be asked.
Any and all thoughts welcome!