Consider for arbitrary $\gamma=\alpha+\beta, \alpha=0,1, \ldots, 0<\beta \leq 1$, the Hölder class $\Lambda_\gamma$ of functions $g(y), y \geq 0$ having finite norm
$$ || g ||_{\gamma}=\sup _{y \geq 0}|g(y)|+\sup _{x, y \geq 0} \frac{\left|g^{(\alpha)}(x)-g^{(\alpha)}(y)\right|}{|x-y|^\beta} . $$
Now consider $g$ to be a bounded between $0$ and $1$, increasing, differentiable and right-continuous function. Does this imply $g \in \Lambda_{\gamma}$?
My thoughts: I think this is not necessarily the case because one can think of $g$ as a cumulative distribution function that has density (thus differentiable) andemphasized text which is clearly right continuous. Then if $g^{\prime}$ is unbounded we cannot find any $\gamma$ such that $|| g ||_{\gamma} < \infty$. I think the above conditions only imply local Lipschitz continuity and since we are dealing with cumulative distribution functions which have density we clearly have uniform continuity as proved here.
Any help is highly appreciated!