  Finite amplitude method based on the deformed relativistic Hartree-Bogoliubov
theory in continuum (DRHBc-FAM) is developed and applied to study isoscalar
giant monopole resonance in exotic nuclei. Validation of the numerical
implementation is examined for $^{208}\textrm{Pb}$. The isoscalar giant
monopole resonances for even-even calcium isotopes from $^{40}\textrm{Ca}$ to
the last bound neutron-rich nucleus $^{80}\textrm{Ca}$ are calculated, and a
good agreement with the available experimental centroid energies is obtained
for $^{40-48}\textrm{Ca}$. For the exotic calcium isotopes, e.g.,
$^{68}\textrm{Ca}$ and $^{80}\textrm{Ca}$, the DRHBc-FAM calculated results are
closer to the energy weighted sum rule than the calculations on the harmonic
oscillator basis, which highlights the advantages of DRHBc-FAM in describing
giant resonances for exotic nuclei. In order to explore the soft monopole mode
in the exotic nuclei, the giant monopole resonance for the deformed exotic
nucleus $^{200}\textrm{Nd}$ is investigated, where the prolate shape and the
oblate shape coexist. A soft monopole mode near 6.0 MeV is found in the prolate
case, and another one near 4.5 MeV is found in the oblate case. The transition
density of the soft monopole mode shows in phase or out-of-phase vibrations
near the surface region, which is generated by quadrupole vibrations.
