  The ring of dual numbers over a ring $R$ is $R[\alpha] = R[x]/(x^2)$, where
$\alpha$ denotes $x+(x^2)$. For any finite commutative ring $R$, we
characterize null polynomials and permutation polynomials on $R[\alpha]$ in
terms of the functions induced by their coordinate polynomials ($f_1,f_2\in
R[x]$, where $f=f_1+\alpha f_2$) and their formal derivatives on $R$.
  We derive explicit formulas for the number of polynomial functions and the
number of polynomial permutations on $\mathbb{Z}_{p^n}[\alpha]$ for $n\le p$
($p$ prime).
