I have a function which lives in $f(x,t)∈L^2(0,T;H^{1/2})∩L^\infty(0,T;L^2)$ for a certain time interval. I also know that $\partial_{t} \ f(x,t)∈L^2(0,T;H^{−1})$. Can I assure that the function lives in $f(x,t)∈C(0,T;L^2)$, i.e., is continuous in time with values in $L^2$?