  In a recent article, the class of functions from the integers to the integers
computable in polynomial time has been characterized using discrete ordinary
differential equations (ODE), also known as finite differences. Doing so, we
pointed out the fundamental role of linear (discrete) ODEs and classical ODE
tools such as changes of variables to capture computability and complexity
measures, or as a tool for programming. In this article, we extend the approach
to a characterization of functions from the integers to the reals computable in
polynomial time in the sense of computable analysis. In particular, we provide
a characterization of such functions in terms of the smallest class of
functions that contains some basic functions, and that is closed by
composition, linear length ODEs, and a natural effective limit schema.
