Reed-Simon's Theorem X.70: strong continuity of a vector-valued function

Question

I'm struggling with understanding the hypothesis of Thm X.70 of Reed and Simon's Methods of mathematical physics. They define $$ C(t,s) = H(t)H(s)^{-1} - I, $$ and state:

Theorem X.70. Let $\mathcal{H}$ be a Hilbert space and let $I$ be an open interval in $\mathbb{R}$. For each $t \in I$, let $H(t)$ be a self-adjoint operator on $\mathcal{H}$ so that $0 \in \rho(H(t))$ and

1. The $H(t)$ have common domain $\mathcal{D}$.
2. For each $\varphi \in \mathcal{H}, (t - s)^{-1} C(t, s)\varphi$ is uniformly strongly continuous and uniformly bounded in $s$ and $t$ for $t \neq s$ lying in any fixed compact subinterval of $I$.
3. For each $\varphi \in \mathcal{H}$, $C(t)\varphi = \lim_{s \nearrow t} (t - s)^{-1}C(t, s)\varphi$ exists uniformly for $t$ in each compact subinterval and $C(t)$ is bounded and strongly continuous in $t$.

Then for all $s \leq t$ in any compact subinterval of $I$ and any $\varphi \in \mathcal{H}$, $$ U(t, s) \varphi = \lim_{k \to \infty} U_k(t, s)\varphi $$ exists uniformly in $s$ and $t$. Further, if $\varphi_s \in \mathcal{D}$, then $\varphi(t) = U(t, s)\varphi_s$ is in $\mathcal{D}$ for all $t$ and satisfies $$ i\frac{d}{dt} \varphi(t) = H(t) \varphi(t), \qquad \varphi(s) = \varphi_s $$ and $\Vert\varphi(t)\Vert = \Vert \varphi_s \Vert $ for all $t \geq s$.

I won't copy the definition of $U_k(t,s)$ since my question is about hypothesis 2 and 3. Initially I thought that 2. means that the operator-valued function $(t-s)^{-1}C(t,s)$ needs to be strongly continuous on $t$ and $s$ instead of the vector-valued function $(t-s)^{-1}C(t,s) \varphi$. That is, for every $\varphi \in \mathcal{H}$, the mappings $s \mapsto (t-s)^{-1}C(t,s) \varphi$ and $t \mapsto (t-s)^{-1}C(t,s) \varphi$ from $\mathbb{R}$ to $\mathcal{H}$ are continuous.

However, I feel I must have misunderstood it. If $(t-s)^{-1}C(t,s)$ is strongly continuous in $s$ and $t$, shouldn't the mappings $t \mapsto \Vert (t-s)^{-1}C(t,s) \varphi \Vert$ and $s \mapsto \Vert (t-s)^{-1}C(t,s) \varphi \Vert$ be continuous? And if that's the case, then $\Vert (t-s)^{-1}C(t,s) \varphi \Vert$ should be uniformly bounded in $s,t$ for $s,t$ in any compact subinterval, doesn't it?

If my interpretation is wrong, I don't know what strong continuity could mean for a function from $\mathbb{R}$ to $\mathcal{H}$; I've tried to find it on Reed-Simon's book and googled it with no success. Can any of you give me some hint or reference?