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

exhibits the same behaviour as the Weierstrass function: There exists a countable dense set

\Gamma \subset R Γ ⊂ R

(the dyadic rationals) such that each Hölder subdifferential

\partial_\alpha H(x) ∂ α H ( x )

is all of

\mathbb R R

for every

x\in\Gamma x ∈ Γ

, while

\partial_\alpha H(x)=\emptyset ∂ α H ( x ) = ∅

for

x\notin \Gamma x ∉ Γ

.