Your argument for the continuity of the electric potential here is circular. If you do not assume a continuous potential V and we consider an integral that intersects the boundary at some point c, then you actually have that the integral of the electric field from b to a is given by V_above(a) - V_above(c) + V_below(c) - V_below(b). The integral expression you've presented here assumes that V_above(c) = V_below(c), which is the continuity condition that you are trying to prove.
Your argument for the continuity of the electric potential here is circular. If you do not assume a continuous potential V and we consider an integral that intersects the boundary at some point c, then you actually have that the integral of the electric field from b to a is given by V_above(a) - V_above(c) + V_below(c) - V_below(b). The integral expression you've presented here assumes that V_above(c) = V_below(c), which is the continuity condition that you are trying to prove.
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