I'm trying to find support for Cauchy’s principal value of $1/x$ given by
$$\left\langle PV(1/x), \phi \right\rangle = PV \int \frac{1}{x}\phi(x)\mathrm{d}x \equiv \lim_{\epsilon \downarrow 0} \int _{|x| > \epsilon } \frac{1}{x} \phi(x)\mathrm{d}x,$$ I use it is distribution, but need support, because distributional derivate is $L^{1}_{loc}$ and i use it.
If $u$ is distribution in $X$ then the support of $u$, denoted $\mathrm{supp}$ $u$, is the set of points in $X$ having no open neighborhood to which the restriction of $u$ is $0$.