Because  $f\in H^1(0,T;H^{-1})$, hence  $f\in C(0,T;H^{-1})$, you may infer that $t\mapsto f(t)$ is continuous into $L^2$ equipped with its weak topology. To prove that it is continous into $L^2$ equipped with its strong (normed) topology, you need that $t\mapsto\|f(t)\|$ be continous ; this is not guaranted by your assumptions, essentially because the spaces $H^{1/2}$ and $H^{-1}$ are not in duality.