We show that for graphene with any finite asymmetry in the on-site energy between the two sublattices (Δ), the optical absorption edge is determined by the Δ. The universal conductance will be broken and the conductance near the band edge varies with frequency as 1/ω2. The onset conductance is σc = 2σ0 = πe2/2h, independent of the size of the band gap. The total integrated optical response is nearly conserved despite of the opening of the band gap. Moreover, near the band edge, there is a change over of the electrical resistivity from temperature independent to a linear temperature dependence.