  We show that a one-dimensional regular continuous 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. By means of a counterexample, we also show that this equivalence fails
for multi-dimensional diffusions. Moreover, for It\^o diffusions we discuss
relations to Cauchy problems.
