Suppose an input voltage changes from 0 to V volts during which a capacitor gets charged fully.Now if the potential is kept constant will the capacitor start discharging or will the charge inside the capacitor remain constant?

Remember that the current through a capacitor is a function of the rate of change of voltage across it:


i = C \frac {dV} {dt}


You can see that if you keep the voltage constant, current flow is 0.

Even if a resistor is connected to it??

even if a resistor is connected to it???

Yep, as long as the voltage across the cap is constant. If that resistor is in parallel with the capacitor you'll get current through the resistor (I = V/R), but not through the capacitor. If the resistor is in series with the cap then there's no current flow - unless the voltage is made to vary.