  We consider Weierstra\ss\ and Takagi-van der Waerden functions with critical
degree of roughness. In this case, the functions have vanishing $p^{\text{th}}$
variation for all $p>1$ but are also nowhere differentiable and hence not of
bounded variation either. We resolve this apparent puzzle by showing that these
functions have finite, nonzero, and linear Wiener--Young $\Phi$-variation along
the sequence of $b$-adic partitions, where $\Phi(x)=x/\sqrt{-\log x}$. For the
Weierstra\ss\ functions, our proof is based on the martingale central limit
theorem (CLT). For the Takagi--van der Waerden functions, we use the CLT for
Markov chains if a certain parameter $b$ is odd, and the standard CLT for $b$
even.
