Why a discrete probability distribution does not converge to a continuous when the number of support points grow to infinity?
Consider a uniform distribution with support in $[0, 1]$, if that is continuous, then pdf in each point in the support would be one, while if the distribution is discrete with (infinitely) many support points the pdf in each point would be zero. Why is that?
Is the concept of pdf inherently different in discrete and continuous cases? I am looking for an intuition about this fundamental question.