  The class of functions from the integers to the integers computable in
polynomial time has been characterized recently using discrete ordinary
differential equations (ODE), also known as finite differences. In the
framework of ordinary differential equations, this is very natural to try to
extend the approach to classes of functions over the reals, and not only over
the integers. Recently, an extension of previous characterization was obtained
for functions from the integers to the reals, but the method used in the proof,
based on the existence of a continuous function from the integers to a suitable
discrete set of reals, cannot extend to functions from the reals to the reals,
as such a function cannot exist for clear topological reasons. In this article,
we prove that this is indeed possible to provide an elegant and simple
algebraic characterization of functions from the reals to the reals: we provide
a characterization of such functions as 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. This is obtained using an
alternative proof technique based on the construction of specific suitable
functions defined recursively, and a barycentric method. Furthermore, we also
extend previous characterizations in several directions: First, we prove that
there is no need of multiplication. We prove a normal form theorem, with a nice
side effect related to formal neural networks. Indeed, given some fixed error
and some polynomial time t(n), our settings produce effectively some neural
network that computes the function over its domain with the given precision,
for any t(n)-polynomial time computable function f .
