  The average fidelity of the teleportation of a coherent state is calculated
for general Gaussian bipartite systems shared by the partners of the protocol,
Alice and Bob. It is considered that the shared Gaussian bipartite modes suffer
independent attenuations before the processing of Alice and Bob. Moreover the
classical communication between the partners can be controlled by a gain not
necessarily unitary. Comparing with the classical fidelity threshold of
measure-and-prepare methods, we establish several genuinely quantum
teleportation conditions which depend on the gain and the local attenuations.
Considering that the gain can be tuned to maximize the bipartite state set able
to genuinely quantum teleportation, a condition for teleportation robust to
local attenuations is found. This condition is demonstrated to be essentially
equivalent to the condition of robust Gaussian bipartite entanglement, obtained
in previous articles, showing that the attenuation robustness is an
entanglement property relevant for characterization and application of
bipartite systems. For the derivation of the robust teleportation conditions,
the Gaussian operations onto the bipartite system are thoroughly studied, so
that the transformations that maintain the fidelity invariant are found. Some
scenarios for different Gaussian bipartite states are presented and discussed.
