Let $t_0 < t\in I$. Then \[ \|F(t) - F(t_0)\| = \Big\| \int_{t_0}^t f(s)\, ds \Big\| \le \int_{t_0}^t \|f(s)\|\, ds. \] Since $s\mapsto \|f(s)\|$ is a real-valued Lebesgue integrable function, we know that $t\mapsto \int_{t_0}^t\|f(s)\|\, ds$ is continuous. Thus, we conclude that $\|F(t) - F(t_0)\| \to 0$ as $t\downarrow t_0$, so $F$ is right continuous at $t_0$. One can show analogously that $F$ is left continuous at $t_0$.