Violating Quantization of Charge? Isn't it the violation of "the quantization of charge" when we use $dQ$ for calculation like integration or derivation?
 A: Generally speaking when we use integration wrt $\mathrm dQ$ we are assuming a continuum limit i.e. the magnitude of the charge is large compared with a charge of a single electron. In practice this is almost always a perfectly safe assumption.
I suppose that in principle we should use a sum over the number of electrons present, but for macroscopic systems it will make no difference. It is, after all, no different to using integration in thermodynamics or fluid mechanics when we assume that the properties of individual atoms/molecules can be ignored.
A: The basic unit of charge is so small by everyday standards that even a small amount of charge contains very many basic units.  Hence for everyday problems in electricity we might as well think of charge as continuous, in the same way we think of water as continuous when in fact it is made of discrete molecules.
What is remarkable is that even in problems involving individual atoms or molecules containing so few charges they must be treated as discrete, the basic laws of electricity remain correct, when these laws were obtained by assuming charge was continuous.
