The following is from Folland's Quantum Field Theory:
Let $\phi$ be a smooth compactly supported function on $\mathbb{R}$. If $\phi(0) \neq 0$, the integral $\int_{-\infty}^\infty |t|^{-1} \phi(t)dt$ is unambiguously divergent. The problem is to find ways of extracting a well-defined "finite part" from it - more precisely, to find distributions $F$ on $\mathbb{R}$ that agree with the function $f(t) = |t|^{-1}$ away from the origin. We observe at the outset that if $F$ is one such distribution, then so is $F + G$ where $G$ is any distribution supported at the origin, that is, any linear combination of the delta-function and its derivatives. We can remove most of this ambiguity by stating the defining condition for $F$ in the following stronger form: we require that $tF(t) = \text{sgn } t$ as distributions; that is if $\phi(t) = t\varphi(t)$, then $\langle F, \phi \rangle = \int (\text{sgn } t) \varphi(t) dt$. Since $t\delta^{(n)}(t) = -n\delta^{(n-1)}(t)$, this reduces the family of allowable $G$'s to the scalar multiples of $\delta$ itself.
Despite reading the above passage multiple times, I understand very little of what its saying or the conclusions it draws. If $F$ is a distribution that agrees with $f(t)$ away from the origin, why is $F + G$ also such a distribution if $G$ is supported at the origin? Does he mean that $G$ is only supported at the origin, or that its support includes the origin? Where did the delta functions come into all of this? Also, we are defining $F$ to be a distribution, so why are we defining $tF(t) = \text{sgn } t$? Should this instead be a new function/distribution?