There exist an integer $n\ge 2$, an [open set](/page/Open%20Set) $U\subset \mathbb R^n$, an exponent $1\le p<\infty$, and a measurable weight $w:U\to(0,\infty)$ with $w<\infty$ $\mathcal L^n$-a.e. such that
is not dense in $W^{1,p}(U,w)$. Here $C^\infty(\overline U)$ denotes the restrictions to $\overline U$ of functions smooth on an open neighbourhood of $\overline U$, so each element of $C^\infty(\overline U)$ has one continuous value at every point of $\overline U$.