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If I have a function which lives in $f(x,t)∈L^2(0,T;H^{1/2})∩L^∞(0,T;L^2)$ 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 funtion lives in $f(x,t)∈C(0,T;L^2)$ this is continious in time which values in $L^2$

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It should be quite direct, but I do not know how to prove it. Thanks in advanced

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 $L^2$?