I was studying a proof and encountered a part where they used the fact that the support of the principal value of $\frac {1}{x}$ that is $Pr (\frac {1}{x})$ is $\mathbb R$.
Please what is the proof of this?
N/B: I know it's quite obvious. In writing I can't put things together.
I had no idea what the question meant until I noticed the tag "distribution-theory". Presumably you're asking about the distrubition $u$ defined by $$\langle u,\phi\rangle = \lim_{a\to0^+}\int_{|t|>a}\frac{\phi(t)}{t}\,dt.$$
It is immensely obvious from the definition that the support of $u$ is $\Bbb R$; so much so that one wonders whether you know the definition (if not you should learn the definition before asking questions about supports of distributions).
By definition, saying that the support of $u$ is $\Bbb R$ means that if $V\subset\Bbb R$ is a nonempty open set then $u$ does not vanish in $V$. By definition, saying $u$ does not vanish in $V$ means that there exists $\phi\in C^\infty_c(V)$ with $\langle u,\phi\rangle\ne0$. And surely that's obvious?
(Ok, even the fact that $C^\infty_c(\Bbb R)\ne\{0\}$ is not a priori obvious, and in fact it's a little surprising at first. But one shows that non-trivial "test functions" exist at the very start of an exposition of the theory of distributions; if one has studied distributions it should be obvious that here wlog $V\subset(0,\infty)$, and there exists $\phi\in C^\infty_c(V)$ with $\phi\ge0$ and $\phi\ne0$; then it's clear that $\langle u,\phi\rangle >0$.)