  The normal distribution is well-known for several results that it is the only
to fulfil. The aim of the present paper is to show that many of these
characterizations actually follow from the fact that the derivative of the
log-density of the normal distribution is the (negative) identity function.
This \emph{a priori} very simple yet surprising observation allows a deeper
understanding of existing characterizations and paves the way to an immediate
extension to a general density $x\mapsto p(x)$ by replacing $-x$ in these
results with $(\log p(x))'$.
