A semigroup $S(t)$ on a Banach space $E$ is a family of bounded linear operators $\{S(t)\}_{t\ge 0}$ with the property that $S(t)S(s)=S(t+s)$ for any $s,t\ge 0$ and that $S(0)=I$. A semigroup is furthermore called strongly continuous if the map $(x,t)\mapsto S(t)x$ is continuous. I was told that this is equivalent of saying $t\to S(t)x$ is continuous for every $x$.
How can I see the equivalence of two ways of defining strong continuity? Could anyone expand what $(x,t)\mapsto S(t)x$ is continuous really mean? Can one show this via the usual strategy of 2-sided continuity? How could this be the same as saying $t\to S(t)x$ is continuous for every $x$?
Appreciate for any helps.