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 $L^2$?



Space time Lesbesgue spaces
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.


**Edits**. Because we already know that $t\mapsto f(t)$ is continuous from $(0,T)$ into $L^2\_w$, we see that the continuity into $L^2$ for the strong topology is equivalent to the continuity of $t\mapsto\|f(t)\|\_L^2$. This is why $f\in L^\infty(0,T;H^{-1/2})$ doesn't help you.


