Singularities of o‐minimal Peano derivatives

Abstract

The notion of Peano differentiability generalizes the differentiability in the usual sense to higher order. Peano differentiable functions have derivatives which are sometimes differentiable or continuous or not even locally bounded. We give a complete characterisation of the sets in which Peano differentiable functions which are definable in an o-minimal expansion of a real closed field are continuously differentiable. Thereby, we also distinguish between several kinds of discontinuities.

Subanalytic blow-$\mathcal {C}^m$ functions

“…The sets of C m -singular points of P m functions have been studied in [9] (see also [7]) for the o-minimal context. Every continuous subanalytic function is locally definable in the o-minimal structure R an consisting of all globally subanalytic sets; cf.…”

Peano differentiable extensions in o-minimal structures

“…The approximation polynomial at some point is uniquely determined by the function, so that for an m times continuously differentiable function, both the approximation polynomial and the Taylor polynomial coincide at any point of the domain. For details about the differences of these differentiability concepts, we refer the reader to [18,Exa 2.5].…”