  It is shown that the spectral radius is continuous on a $C^*$-algebra if and
only if the $C^*$-algebra is type I. This answers a question of V. Shulman and
Yu.~Turovskii [10]. It is shown also that the closure of nilpotents in a
$C^*$-algebra contains an element with non-zero spectrum if and only if the
$C^*$-algebra is not type I.
