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We first prove that the property of strict monotonicity of a Köthe space $(E,\parallel .{\parallel }_E)$ and/or of its Köthe dual $(E^{\text{'}},\parallel .{\parallel }_{E^{\text{'}}})$ can be used successfully to compare the supports of $x\in E\setminus \left\{\theta \right\}$ and $y\in S\left(E^{\text{'}}\right)$, where $={\parallel x\parallel }_E$. Next we prove that any element $x\in S_{+}\left(E\right)$ with $\mu (T\setminus suppx)=0$ is a point of order smoothness in $E$, whenever $E$ is an order continuous Köthe space. Finally, we present formulas for the characteristic of monotonicity of Orlicz function spaces endowed with the Orlicz norm in the case when the generating Orlicz function does not satisfy suitable ${\Delta }_2$-condition...