discontinuous galerkin method double-valued boundary values

Question

when looking at a simple example for a DG-Method, lets make the domain go from 0 to 2, with one point in the middle: \begin{align} -u''=f \end{align} we would multiply with a smooth function and integrate by parts: \begin{align} \int_0^2 -u''v=\int_0^1 - u''v+\int_1^2 -u''v = \\-u'(1)v(1)+u'(0)v(0)+\int_0^1 u'v'-u'(2)v(2)+u'(1)v(1)+\int_1^2 u'v' \end{align} Now, how does one handle that values at (here only at $x=1$) the ends of an interval are potentially double valued, due to discontinuities? In my papers there is a lot about jumps and mean values, but I do not get how this can be included into the above equations. Any ideas?