I know that voltage of a capacitor is continuous. However, I want to learn that; can it be a jump in the capacitor voltage? Also, if it is, how it can be happened?
Classically, to charge a capacitor requires current. To charge a capacitor in zero time requires infinite current, which is not possible. I could imagine that if the charging were very fast, it might be reasonable to approximate it as an instantaneous jump in voltage.
But charge is quantized. I wonder what happens in the quantum mechanical case.
charge on a capacitor is given by q = CV...there is superficially no time here ...BUT
atyy's explanation is a good one: charge is the result of the flow of electrons which have have finite, not instantaneous speed...in fact charge in coulombs is given by q=it...amps times time...so in this formulation you can see...charging takes some "t"...
in fact a step function (square wave) voltage will charge it really,really,fast...but not instantaneously...