  By extending the idea of a difference operator with a fixed step to
varying-steps difference operators, we have established a difference Nevanlinna
theory for meromorphic functions with the steps tending to zero (vanishing
period) and a difference Nevanlinna theory for finite order meromorphic
functions with the steps tending to infinity (infinite period) in this paper.
We can recover the classical little Picard theorem from the vanishing period
theory, but we require additional finite order growth restriction for
meromorphic functions from the infinite period theory. Then we give some
applications of our theories to exhibit connections between discrete equations
and and their continuous analogues.
