Why does the expression of current as rate at which charges flow through a given surface makes sense? We know that charges are quantised, ie a charge on a body q can be expressed as

q = ne

where e is the modulus of charge on an electron and n∈N.
But when providing an expression for current, my textbook says 

I=dq/dt

Mathematically, this equation is not supposed to make sense since charge on a body is not a continuous function of time(any non-continuous function will not be differentiable at the points where the function is not continuous).
Yet, we accept this equation and go on with the subject of electromagnetism. Am I missing something?
 A: It's the same reason we measure water in gallons/liters instead of counting the number of molecules. The individual units (electrons or water molecules) are so small and so numerous that treating their flow as if they were a fluid results in immeasurably tiny errors. In a wire with an amp of current flowing through, $6\cdot10^{18}$ electrons are passing through every second. Throwing in a partial electron charge means nothing next to this large a quantity. A single electron is such a small quantity of charge that is almost physically embodies the infintesimal $dq$. The benefit of this continuous approximation is that continuous functions are usually easier to work with mathematically than discrete functions.
A: The discrete nature of the charge carrier (the electron) manifests itself in shot noise (random fluctuation of electric current) which is not normally noticed because the number of electrons conveying the current is so large that the statistical fluctuations in the current are very, very small compared with the (average) current.
A: You should try think about the rate of change of this electron passing from one point to another.  Then try to visualize the cross section of a wire and the rate of change of many electrons through this area.  
Noise is what you see on an oscilloscope when the wave isn't showing perfectly.  Noise is the little ripples.  
