**Question:**  Let $f: [0, 1] \to \mathbb R$ be a continuous, nowhere differentiable function. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ in sup norm. Is it true that for all $1 \leq p \leq \infty$, we have $\|u_n\|_{W^{1, p}} \to \infty?$