I've always been told to use discrete functions with discrete data and continuous functions with continuous data. This especially comes to light when talking stats and people emphasize the difference between discrete and continuous probability distributions.
For example the Pareto distribution and its discrete counter part Zipf's law or sometimes called the Zeta Distribution.
Now if I have data that is discrete and seems to follow this 80-20 rule suggested by the above models, what is wrong with simply choosing to use the Pareto distribution but only looking at he values at whole numbers?
I believe that one of the reasons is that $$\sum_{x \in S} P(X=x)=1$$ for discrete, but for continuous it is $$\int_\mathbb{R}P(X=x)dx=1$$. So, if you look at only integers in the continuous case, they do not form a probability distribution by themselves, since you "miss" the in between values. Of course, you can use that trick to approximate the discrete distribution, but it won't be exactly the same.