How is the fundamental theorem of calculus used in a proof about the mild representation formula for the solution to an abstract ODE that I'm reading? Let


*

*$E$ be a $\mathbb R$-Banach space

*$S:[0,\infty)\to E$ be a $C^0$-semigroup

*$T>0$

*$f\in C^0([0,T],E)$


Since $S$ is a $C^0$-semigroup, $$[0,\infty)\to E\;,\;\;\;t\mapsto S(t)x\tag1$$ is continuous for all $x\in E$. However, unless $S$ is even uniformly continous, this shouldn't imply the continuity of $$[0,t]\to E\;,\;\;\;s\mapsto S(t-s)f(s)\tag2$$ for all $t\in(0,T]$.
However, how is then the fundamental theorem of calculus used in equation (12.28) of the book An Introduction to Partial Differential Equations by Michael Renardy and Robert C. Rogers?

Excerpt of the mentioned book (the authors use the symbol $T$ for both the maximal time and the semigroup $S$):


I think everything is okay. Because $S$ is uniformly bounded in operator norm on any finite interval $[0,T]$, then
\begin{align}
         & \|S(t-s)f(s)-S(t-s')f(s')\| \\
         &= \|\{S(t-s)-S(t-s')\}f(s)+S(t-s')\{f(s)-f(s')\}\| \\
         &\le \|\{S(t-s)-S(t-s')\}f(s)\|+M\|f(s)-f(s')\|.
\end{align}
Hence,
$$
         \lim_{s'\rightarrow s}S(t-s')f(s')=S(t-s)f(s).
$$
