Let us consider $f\in {L^2(\mathbb{R}^N)}$ such that there exists the weak derivative $\partial_{x_1}(|f|^2)\in {L^2(\mathbb{R}^N)}$, can we ensure that $\displaystyle \int_{\mathbb{R}^N}\partial_{x_1}(|f|^2)=0$?
I know that there is a version of Fundamental Calculus Theorem for weak derivative which says that (fixing $x_2,\dots,x_N$) we can find a continuous function, $\bar{f}$, such that $\bar{f}=|f|^2$ a.e. and $\displaystyle \int_{\mathbb{R}^N}\partial_{x_1}(|f|^2)=\displaystyle \int_{\mathbb{R}^{N-1}} \left(\lim_{x_1\to+\infty} \bar f-\lim_{x_1\to-\infty} \bar f\right)$. So, the proof will be complete if I prove that$\displaystyle\lim_{x_1\to\pm\infty} \bar f=0$.
Could someone help me with this last step?