I know that the Heaviside function ($H(x) = 1$ if $x\ge 0$ and $0$ if $x<0$) has derivative $\delta_o$ on $(-1,1)$ (in sense of distribution). Now I want to find the distributional derivative of $\chi_{\mathbb{Q}}$ (characteristic function of rational numbers).
\begin{equation} \int_{-1}^1 \chi_{\mathbb{Q}}' \, \phi \,dx = - \int_{-1}^1 \chi_{\mathbb{Q}}\, \phi'\, dx \end{equation}
Now on the right I have to integrate the characteristic function, so it is the zero function. Is it correct to say that the (distributional) derivative of $\chi_{\mathbb{Q}}$ is the zero function?