For some $phi$-sequences $phi_1$, $phi_2$ and $phi_3$, and $kappa$-function $kappa_1$, $kappa_2$ and $kappa_3$ with $kappa_1^{-1}(x)kappa_2^{-1}(x);{geq};kappa_3^{-1}(x)$ for $x;{geq};0$, the Luxemburg norm is lower semi-continuous on ${kappa}{phi}BV_0$, and some specialized equivalent conditions are considered.