I have write a little argument that seems to show that every function $f \in L^2(\Omega)$ admits a weak derivative:
Let $f \in L^2(\Omega)$. We define the functional on $H^1(\Omega) = W^{1, 2}(\Omega)$ $$\mathcal F: H^1(\Omega) \to \mathbb R: u \mapsto -\int_\Omega f\partial_iu$$ which is continuous because $$|\mathcal F(u)| \le \|f\|_2 \|u\|_{1, 2}.$$ By Riesz representation theorem, there is $\tilde{f}\in H^1(\Omega)$ such that $$\int_\Omega \tilde{f}u = -\int_\Omega f\partial_iu, \quad \forall u \in H^1(\Omega).$$ As $C^\infty_0(\Omega) \subset H^1(\Omega)$, this equality is true for all $u \in C_0^\infty(\Omega)$. Moreover, as $\tilde{f} \in H^1(\Omega)$, in particular $\tilde{f} \in L^2(\Omega)$ so that $f$ admits a weak derivative given by $\tilde{f}$.
This argument has to be wrong because not all functions $L^2(\Omega)$ have a weak derivative but I really don't see where is my error. Could one have you help me with this ?