How is that circuit wont be broken even there is an air gap in between the capacitor??

air acts as dielectric material it transmits electrical effects without actually conducting electricity.dats y circuit is not broken

The electrons don''t flow through the air gap. When one plate is charged with a particular amount of charge, say positive, it will induce charge on the other plate though it is kept at a distance. This is because both the plates are connected to the opp terminals of a battery and hence, if charge(-ve) is removed from one plate( the positive one), then to maintain the flow through the battery, it must add the -ve charge to the other plate, thereby making it negative! 

In AC circuit capacitor offers resistance to current as aresistor would do in DC. But for DC there exists a current in between the plates called displacement current (it exists everywhere in fact)

I_d = ε₀x dΦ_e/dt

Modified amperes law = integral of B.dl = mu0 ( I + I_d)

there exists a magnetic field between the plates responsible for transfer of charges from one plate to another and hence I= 0

but I_d is not 0. At every I_d = I