Why is the initial current in this capacitor not continuous?
At \$t=0^{-}\$ the current in the capacitor is zero, since it's an open circuit when only DC voltages are applied in the circuit.

At \$t=0\$, \$V_2\$ comes in, and so does the current through the capacitor, which is now \$\frac{V_2 - V_1}{R_2}\$ since at \$t=0\$ no current flows through the inductor branch.

Finally, remember that the capacitor current can change instantaneously, it's its voltage that cannot.