Abstract We study [ φ t , X ] , the maximal space of strong continuity for a semigroup of composition operators induced by a semigroup { φ t } t ≥ 0 of analytic self-maps of the unit disk, when X is BMOA, H ∞ or the disk algebra. In particular, we show that [ φ t , BMOA ] ≠ BMOA for all nontrivial semigroups. We also prove, for every semigroup { φ t } t ≥ 0 , that lim t → 0 + ⁡ φ t ( z ) = z not just pointwise, but in H ∞ norm. This provides a unified proof of known results about [ φ t , X ] when X ∈ { H p , A p , B 0 , VMOA } .