Automatic time continuity of positive matrix and operator semigroups

We consider a matrix semigroup $T: [0,\infty) \to \mathbb{R}^{d \times d}$ without assuming any measurability properties and show that, if $T$ is bounded close to $0$ and $T(t) \ge 0$ entrywise for all $t$, then $T$ is continuous. This complements classical results for the scalar-valued case. We also prove an analogous result if $T$ takes values in the positive operators over a sequence space.