Calculus on a non-Archimedean field extension of the real numbers: inverse function theorem, intermediate value theorem and mean value theorem

In this paper, we introduce the concept of weakly locally uniformly differentiable functions (WLUD) on N \mathcal {N} , a non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. We show that WLUD functions are C 1 C^1 and they form an N \mathcal {N} -algebra that is closed under composition and contains all polynomial functions. We formulate and prove a version of the inverse function theorem as well as a local intermediate value theorem for these functions. Then we generalize the WLUD concept to higher orders of differentiability and study WLUD n ^n functions at a point or on a subset of N \mathcal {N} . In particular, we study the properties of WLUD 2 ^2 functions and we formulate and prove a local mean value theorem for such functions.