  Let $(X,d,\mu)$ be a doubling metric measure space. We consider the behaviour
of the fractional maximal function $M^\alpha$ for $0\leq \alpha<Q$, where $Q$
is the doubling dimension, acting on functions of bounded mean oscillation
(BMO) and vanishing mean oscillation ($VMO$). For $\alpha>0$, we additionally
assume that the space is bounded. We show that $M^\alpha$ is bounded from $BMO$
to $BLO$, a subclass of $BMO$, and maps $VMO$ to itself when $\mu$ has the
annular decay property. We also show by means of examples that the action of
$M^\alpha$ is not continuous on these function spaces.
