Why do we assume differential coefficients of number of molecules? In many portions of physics (like Maxwell's velocity distribution law) we assume statements like- 

Number of molecules having velocity between $c$ to $c+dc$ is $dn$.

But number of molecules $n$ is always an integer and thus $dn$ i.e. differential coefficient of $n$ can't be determined due to infinite discontinuity between two integers. So why do we write & calculate using $dn$?
 A: I think the other answers are more easily summarized as:
A continuous function is easier to integrate (or differentiate) and provides an answer which is more than accurate enough for any use.
This is similar to replacing binomial distributions with continuous distributions for large populations. 
A: We can consider this to be the mean number of molecules, or the expectation value. There are almost always random fluctations to the numbers of molecules in any given interval -- say if 50% of the time we expect 2, and the other 50% 3, the expectation value or mean value would be 2.5 -- even though we can never see 2.5 molecules. In this way, the distribution becomes continuous, and using d(n) is makes more sense. 
A: The problem is that in order to work with distributions of velocities / densities etc, we need to consider "every possible value". There may well not be, at any given moment, any molecules with that precise value - in fact, if you specify the value to enough precision, there will never be a single molecule that has that value.
And so we move from the realm of "discrete" to "continuous". If I have an object with speed that uniformly changes between 1 m/s and 2 m/s, what is the probability I will find it has a speed of 1.5 m/s? It's zero. But might find an object with a velocity between 1.45 m/s and 1.55 m/s - the probability would be 10 % (since 1.55 - 1.45 = 0.1, which is 10% of the total range).
If I look for an object with that velocity, I will either find none (90% of the time) or one (10% of the time). I express that by saying the probability is 10%. I will never find 0.1 particle there...
Mathematically, we can describe the probability density function as some function that indicates the probability of finding something "in an interval" (never "at a value"). That interval can be finite, as it was in my example above (0.1 m/s), but we can mathematically make the size of the interval "go to zero" by writing $dx$ as the size of the interval.
As the interval becomes smaller, the probability becomes smaller - but still, the sum of all the probabilities (the integral) can be finite.
