Let $U\subset \Bbb{R}^n$ be the bounded domain.
Is there some example such that $f\in C^1(U)$ but not $W^{1,p}(U)$.We know for bounded $U$ we have $$C^1(\overline{U})\subset W^{1,p}(U)$$
Intuitivly the compact set give above is used to prevent the "blowing up" of $f$ near boundary,is there some concrete example for example in 1-dimension?