Let $f$ be a concave function on $N_{\mathbb{R}}$, $\lambda>0$, $u_{0}\in N_{\mathbb{R}}$ and $x_{0}\in M_{\mathbb{R}}$. Then

1. (1)

   ${\operatorname{dom}}(\lambda f)={\operatorname{dom}}(f)$, $\operatorname{stab}(\lambda f)=\lambda\operatorname{stab}(f)$ and $(\lambda f)^{\vee}=f^{\vee}\lambda$;

2. (2)

   ${\operatorname{dom}}(f\lambda)=\lambda{\operatorname{dom}}(f)$, $\operatorname{stab}(f\lambda)=\operatorname{stab}(f)$ and $(f\lambda)^{\vee}=\lambda f^{\vee}$;

3. (3)

   ${\operatorname{dom}}(\tau_{u_{0}}f)={\operatorname{dom}}(f)+u_{0}$, $\operatorname{stab}(\tau_{u_{0}}f)=\operatorname{stab}(f)$ and $(\tau_{u_{0}}f)^{\vee}=f^{\vee}+u_{0}$;

4. (4)

   ${\operatorname{dom}}(f+x_{0})={\operatorname{dom}}(f)$, $\operatorname{stab}(f+x_{0})=\operatorname{stab}(f)+x_{0}$ and $(f+x_{0})^{\vee}=\tau_{x_{0}}f^{\vee}$.