A new look at the trailing zeroes on $N!$

Let us denote by $Z_b(n)$ the number of trailing zeroes in the base $b$ expansion of $n!$. In this paper we study with some detail the behavior of the function $Z_b$. In particular, since $Z_b$ is non-decreasing, we will characterize the points where it increases and we will compute the amplitude of the jump in each of such points. In passing, we will study some asymptotic aspects and we will give families of integers that do not belong to the image of $Z_b$.