  Mixed monotone systems form an important class of nonlinear systems that have
recently received attention in the abstraction-based control design area.
Slightly different definitions exist in the literature, and it remains a
challenge to verify mixed monotonicity of a system in general. In this paper,
we first clarify the relation between different existing definitions of mixed
monotone systems, and then give two sufficient conditions for mixed monotone
functions defined on Euclidean space. These sufficient conditions are more
general than the ones from the existing control literature, and they suggest
that mixed monotonicity is a very generic property. Some discussions are
provided on the computational usefulness of the proposed sufficient conditions.
