On the almost decrease of a subexponential density

For a subexponential density, so far, there has been no positive conclusion or counter example to show whether it is almost decreasing. In this paper, a subexponential density supported on $\mathbb{R}^+\cup\{0\}$ without the almost decrease is constructed by a little skillful method. The density is a positive piecewise linear function with a more normal shape. Correspondingly, there exists a local subexponential distribution which is not locally almost decreasing. Based on an example of Cline \cite{C1986}, some similar results are also obtained for the long-tailed density excluding the subexponential density and the local long-tailed distribution excluding the local subexponential distribution. Finally, the paper shows that, for the local subexponentiality of a distribution supported on $\mathbb{R}$, the local almost decreasing condition is necessary in some sense.