Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly.
Questions:
i) Is it true that if $f$ is nowhere differentiable, then $\|u_n\|_{W^{1, 1}} \to \infty?$
ii) Is it true that if $f$ fails to be absolutely continuous, then $\|u_n\|_{W^{1, p}} \to \infty$ for $p > 1$?