Why is it possible to find electrostatic energy of a conductor by integration of $int_{space} vec{E}^2 dV$?

In Griffith’s Introduction to Electrostatics, International 4th ed, pg-94 these two equations are given for calculating energy of continous charge distributions:

$$ W= frac12 int rho V d tau tag{1}$$

And after some simplifications, this other equation is given $$ W= frac{epsilon_o}{2} int_{whole space} vec{E}^2 d tau tag{2}$$

And, there are also some exercises to find the energy of conductors using equation (2) but after some careful thought, I realize that in practice, equation (2) may have to integrate over a boundary. This is problem because the component of electric field normal to a surface is discontinuous by an amount $frac{sigma}{epsilon}$. Now, since we have a discontinuity at the boundary, how can is the expression integrable?(*)

As far as I learned, discontinuous functions such as $ frac{1}{x}$ etc are not integrable on a set containing their discontinuity.

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On some reflection, I started realize voltage of a conductor maybe suffering from the same problem because when we integrate from infinity to a point inside a conductor, there is a sudden discontinuity of the electric field at the boundary again. Hence, (1) also suffers the same problem.