I am trying to find an example of a function $u\in L_2((-2,2))$ such that $||\delta_h(u)||_{L_1((-1,1))}$ is uniformly bounded in $0<|h|<1/2$ but $u'$ is not in $L_1((-1,1))$. Where $\delta_h(u)(x) = \frac{u(x+h)-u(x)}{h}$.
My idea is to use a Lipschitz continuous function that is not everywhere differentiable, but all the examples I can come up to have derivatives in $L_1((-1,1))$. If someone told me whether I am in the right track or should look for another kind of functions I would greatly appreciate it.