Voltage and current of a capacitor while discharging
At the instant a load is connected to a supply like a capacitor, the load voltage and the supply voltage become equal, regardless of the load. The current then evolves according to what the load is. If the load has some inductance, then the current rises slowly. If the load is pure resistance, and the supply is ideal, and everything is small, then the current rises instantly.

Don't get too hung up on the impossibility of the current rising instantly into a resistive load, it never does. The source and load will always have some inductance, and the circuit is never 'small', there will always be transmission line effects. If you model ideal components, then you will get ideal instant behaviour. If you model sources, connections and loads with inductance and physical length, then you'll see more realistic finite things happening.