  We give several new characterizations of completely monotone functions and
Bernstein functions via two approaches: the first one is driven algebraically
via elementary preserving mappings and the second one is developed in terms of
the behavior of their restriction on the set of non-negative integers. We give
a complete answer to the following question: Can we affirm that a function is
completely monotone (resp. a Bernstein function) if we know that the sequence
formed by its restriction on the integers is completely monotone (resp.
alternating)? This approach constitutes a kind of converse of Hausdorff's
moment characterization theorem in the context of completely monotone
sequences.
