On the total curvatures of a tame function

TL;DR

This paper investigates the continuity properties of total curvature functions associated with definable, sufficiently differentiable functions, establishing that these curvature functions have only finitely many discontinuities.

Contribution

It proves that the total curvature and total absolute curvature functions of definable functions have at most finitely many discontinuities.

Findings

Total curvature functions are finitely discontinuous.

Total absolute curvature functions are finitely discontinuous.

Provides a finiteness result for curvature discontinuities in definable functions.

Abstract

Given a definable function f, enough differentiable, we study the continuity of the total curvature function t --> K(t), total curvature of the level {f=t}, and the total absolute curvature function t-->|K| (t), total absolute curvature of the level {f=t}. We show they admits at most finitely many discontinuities.