  We show that a one-dimensional regular continuous strong Markov process \(X\)
with scale function \(s\) is a Feller-Dynkin process precisely if the space
transformed process \(s (X)\) is a martingale when stopped at the boundaries of
its state space. As a consequence, the Feller-Dynkin and the martingale
property are equivalent for regular diffusions on natural scale with open state
space. Furthermore, for It\^o diffusions we discuss relations to existence and
uniqueness properties of Cauchy problems, and we identify the infinitesimal
generator.
