A basic question about $W^{1,1}(\Omega)$

Question

Let $\Omega$ be an open interval $(-1,1)$. Does there exist $u$ in $L^1(\Omega)$ such that $u$ is not in $W^{1,1}(\Omega)$?

Let $I=(a,b)$ ba an open interval, possibly unbounded.

$W^{1,1}(I)=\left\{ u\in L^{1}\left(I\right):\exists g\in L^{1}\left(I\right)\mathrm{such\; that}\int_{I}u\varphi'=-\int_{I}g\varphi\;\forall\varphi\in C_{c}^{1}\left(I\right)\right\} $

I know that such $u$ exists but I can't write it precisely. Can anyone help me?

Thanks in advanced.

Answer 1

$u(x)=\begin{cases} 1 & x\in\left(0,1\right)\\ -1 & x\in\left(-1,0\right) \end{cases}$