Let $U_t$ be a one parameter subgroup of normal bounded operators on a complex Hilbert space $H$. For each $t\in \mathbb{R}$, $U_t$ defines on the Borel subsets of $\mathbb{C}$ a projection valued measure $P_t:\mathcal{B}(\mathbb{C}) \to B(H)$.
Now let $f \in C(\mathbb{C})$ is the map $F:\mathbb{R} \to B(H)$ defined by: $$ \int_{\mathbb{C}}f(s)dP_t(s)$$ Is it continuous.
All of this is very new to me so I hope that the question makes sense.
If $f$ is the identity, then $F$ is just the original unitary group. Thus in general, the best continuity of $F$ one can hope for is that of $(U_t)$. If as usual, $(U_t)$ is strongly continuous, then $F$ is strongly continuous as well. This can be proven by approximating $F$ uniformly by polynomials in $z$ and $\bar z$ (see Takesaki. Theory of Operator Algebras I, Lemma II.4.3).