Derivation of anomalous commutators of currents in Fradkin's book I am trying to understand the derivation of the anomalous commutators of the left- (and right)-moving currents in Fradkin's book (see e.g. here). I am not sure I understand how (6.71) leads to (6.72).
My understanding is that we can work out identities with the Delta-Distribution for test functions. Taking $\epsilon,\epsilon'=0$ we have
$$ \int \left(\frac{\delta(x-x')}{x-x'} - \frac{\delta(x-x')}{x'-x}\right) f(x) dx = \int \frac{\delta(x) f(x+x')}{x} - \frac{\delta(x)f(x'-x)}{x}  dx$$
which almost looks like $-f(x)\partial_x \delta(x-x') \simeq \partial_x f(x)|_{x=x'} = \lim_{h\to0} \frac{f(x'+h)-f(x'-h)}{2 h}$, but I seem to be missing a crucial factor of $2$ here. I am not sure if I am missing something more formal here. (I am actually trying to get a consistent convention of bosonization identities for myself, so it seems rather crucial.
 A: Note that these equations are (5.244) and (5.245) in the book version, compared to the online notes. Luckily I still have my own notes scribbled in this section of my book copy. Here it is:
$$
\begin{split}
[j_+(x), j_+(x')] &= \lim_{\epsilon, \epsilon' \rightarrow 0} \left( \frac{i \delta(x' - x + \epsilon' + \epsilon)}{2\pi(x - x' + \epsilon + \epsilon')} - \frac{i \delta(x - x' + \epsilon' + \epsilon)}{2\pi(x' - x + \epsilon + \epsilon')} \right)\\
&= \lim_{\epsilon, \epsilon' \rightarrow 0} \left( \frac{i \delta(x' - x + \epsilon' + \epsilon)}{2\pi(\epsilon' +\epsilon + \epsilon + \epsilon')} - \frac{i \delta(x - x' + \epsilon' + \epsilon)}{2\pi(\epsilon + \epsilon' + \epsilon + \epsilon')} \right)\\
&= \lim_{\epsilon \rightarrow 0} \left(\frac{i \delta(x' - x + \epsilon)}{4\pi \epsilon} - \frac{i \delta(x-x'+\epsilon)}{4\pi \epsilon} \right) \\
&= -\frac{i}{2\pi} \partial_x \delta(x-x'). 
\end{split}
$$
Essentially, all I did was use the delta functions to replace $x-x'$ in the denominator with epsilons, which gives the desired factor of two.
