  We present a new method for constructing $C_0$-semigroups for which
properties of the resolvent of the generator and continuity properties of the
semigroup in the operator-norm topology are controlled simultaneously. It
allows us to show that a) there exists a $C_0$-semigroup which is continuous in
the operator-norm topology for no $t \in [0,1]$ such that the resolvent of its
generator has a logarithmic decay at infinity along vertical lines; b) there
exists a $C_0$-semigroup which is continuous in the operator-norm topology for
no $t \in \mathbb R_+$ such that the resolvent of its generator has a decay
along vertical lines arbitrarily close to a logarithmic one. These examples
rule out any possibility of characterizing norm-continuity of semigroups on
arbitrary Banach spaces in terms of resolvent-norm decay on vertical lines.
