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