  We consider the decreasing and the increasing $r$-excessive functions
$\varphi_r$ and $\psi_r$ that are associated with a one-dimensional
conservative regular continuous strong Markov process $X$ with values in an
interval with endpoints $\alpha < \beta$. We prove that the $r$-excessive local
martingale $\bigl( e^{-r (t \wedge T_\alpha)} \varphi_r (X_{t \wedge T_\alpha})
\bigr)$ $\bigl($resp., $\bigl( e^{-r (t \wedge T_\beta)} \psi_r (X_{t \wedge
T_\beta}) \bigr) \bigr)$ is a strict local martingale if the boundary point
$\alpha$ (resp., $\beta$) is inaccessible and entrance, and a martingale
otherwise.
