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?

Question

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 $S$):

Answer 1

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, $$ \lim_{s'\rightarrow s}S(t-s')f(s')=S(t-s)f(s). $$