Subdifferential Analysis of the Van der Waerden Function

A concise and direct proof is given that Hölder subdifferentials of the (continuous but nowhere differentiable) Van der Waerden function $H(\cdot)$ exhibits the same behaviour as the Weierstrass function: There exists a countable dense set $\Gamma \subset R$ (the dyadic rationals) such that each Hölder subdifferential $\partial_\alpha H(x)$ is all of $\mathbb R$ for every $x\in\Gamma$, while $\partial_\alpha H(x)=\emptyset$ for $x\notin \Gamma$.