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Is there any function in $H^1(\Omega)$ but not in $C(\Omega)$ other than singular functions? (Given the dimension $N\geq2$)

For an example, assume $N\geq 3$ and let $\Omega:=\{x\in\mathbb{R}^N;|x|<1\}$, then define a function almost everywhere in $\Omega$ as: $$ u(x):=|x|^{-\lambda},\quad x\neq0 $$ where $0<\lambda<(N-2)/2$. Then $u\in H^1(\Omega)$ and $u\notin C(\Omega)$.

However, could I find another example such that the function is also not singular at some points?

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    $\begingroup$ Depends on exactly what you mean by "singular". $\endgroup$ – David C. Ullrich Nov 12 '17 at 13:56
  • $\begingroup$ I mean I need some examples look like much more discontinuous. Just like a piecewise constant function, but I know it's not H^1. $\endgroup$ – whereamI Nov 12 '17 at 13:58
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    $\begingroup$ If I remember correctly, $\sin(u)$ with your $u$ should do the job. The sine converts the singularity into a discontinuity. $\endgroup$ – gerw Nov 13 '17 at 7:09

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