I have a divergent integral of the following form:
$$I = - \int_\mathbb{R} d\tau_3 \int_\mathbb{R} d\tau_4 \int_\mathbb{R} d\tau_5 \int_{\mathbb{R}^4} d^4 x_6\ \text{sgn} (\tau_{34})\ \text{sgn} (\tau_{35})\ \text{sgn} (\tau_{45}) \left( \partial_{\tau_2} - \partial_{\tau_5} \right) I_{26} I_{46} I_{56}, \tag{1}$$
where I have defined:
$$x_i := (x^1_i, x^2_i, x^3_i, \tau_i), \tag{2.a}$$ $$\tau_{ij} := \tau_i - \tau_j, \tag{2.b}$$ $$I_{ij} := \frac{1}{(x_i - x_j)^2}. \tag{2.c}$$
I would like to integrate by parts such that I can rewrite eq. $(1)$ as:
$$I \overset{?}{=} + \int_\mathbb{R} d\tau_3 \int_\mathbb{R} d\tau_4 \int_\mathbb{R} d\tau_5 \int_{\mathbb{R}^4} d^4 x_6\ I_{26} I_{46} I_{56} \left( \partial_{\tau_4} + 2 \partial_{\tau_5} \right) \left( \text{sgn} (\tau_{34})\ \text{sgn} (\tau_{35})\ \text{sgn} (\tau_{45}) \right). \tag{3}$$
The reason I would like to do so is that I can then separate the finite and the divergent parts of the integral.
To obtain eq. $(3)$, I integrate by parts twice, and what I would like to know is precisely if those steps are allowed. The ibp's are the following:
$$\int_\mathbb{R} d \tau_6\ \partial_{\tau_2} I_{26} I_{46} I_{56} \overset{?}{=} - \int_\mathbb{R} d \tau_6\ I_{26} \left( \partial_{\tau_4} + \partial_{\tau_5} \right) \left( I_{46} I_{56} \right) \tag{4.a}$$
$$\int_\mathbb{R} d\tau_4\ \text{sgn} (\tau_{34})\ \text{sgn} (\tau_{35})\ \text{sgn} (\tau_{45})\ \partial_{\tau_4} I_{46} \overset{?}{=} - \int_\mathbb{R} d\tau_4\ I_{46}\ \partial_{\tau_4} (\text{sgn} (\tau_{34})\ \text{sgn} (\tau_{35})\ \text{sgn} (\tau_{45})) \tag{4.b}$$
In the first equality I have used:
$$\partial_{\tau_i} I_{ij} = - \partial_{\tau_j} I_{ij}. \tag{5}$$
So are $(4.a)$ and $(4.b)$ correct? I am unsure, because both the $I$-functions as well as the sign-functions are not continuous on $\mathbb{R}$ (but they are piecewise continuous).