Gauss and the identity function -- a tale of characterizations of the normal distribution

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 $(\log p(x))'$.