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 .

翻译：从整数到多元时间可计算整数的函数类别最近使用离散的普通差分方程式(ODE)来定性。在普通差分方程式(ODE)的框架中,试图将方法扩大到真实函数的类别,而不仅仅是整数的整数。最近,从整数到真实时间的函数(ODE),从整数到可计算整数的函数类别,但根据从整数到合适的真实时间组的连续函数(ODE),不能延伸到从真实的普通差分方程式(ODE),也称为有限的差异。在普通差异方程式的框架中,这种函数无法有效地存在。在普通差分方程式的框架中,我们证明这确实可以提供从真实到真实的函数的优美和简单的代数。我们为包含一些基本功能的最小的函数类别提供了先前的定性,并且由于给定的构成、线性值的直径直值和自然有效限,这是用基于正常的正数网络的可变数技术获得的。我们用某种正数的正数的正值来证明我们之前的正数的正数函数。我们需要一个特定的正数分析。我们之前的正数的正数。我们需要的某的正数的正数的正数。我们需要的正数。