Here is the definition of a continuous semigroup
Let $C$ be a subset of a Banach space $X$. A semigroup on $C$ is a group $\{S(t):t\geq 0\}$ of a self-maps defined on the subset $C$ which satisfies the following properties
- $S(0)=I,$
- $S(t+s)x=S(t)S(s)x, \forall t,s\geq 0,\; x\in C.$
A semi-group $S(t)$ is said to be continuous if the mapping $h:[0, \infty) \times C \rightarrow C$ defined by $h(t,x)=S(t)x$ is continuous when the subset $C$ carries the norm topology of the element $x$.
What we mean by saying "$h(t,x)=S(t)x$ is continuous when the subset $C$ carries the norm topology of the element $x$"?