How can the current change suddenly in an RC circuit?
I can understand that the voltage can change immediately, but the current?


When the capacitor is charged there is 12 V on it.
When you switch to the discharge resistor you have 12 V across 500 Ω. You should expect an immediate 24 mA to flow and this will decrease as explained by the RC discharge curve.
When the capacitor is full discharged it will (initally) appear like a short-circuit to ground. As soon as you switch on the charge 24 mA will flow into the capacitor.


Shouldn't it have a smooth curve?

No, the simulator is correct.