Nowhere monotone Riemann integrable derivative without local extrema

TL;DR

This paper constructs specific examples of differentiable functions with Riemann integrable derivatives that are nowhere monotone and lack local extrema, challenging common assumptions about derivatives and extrema.

Contribution

It introduces novel functions demonstrating that derivatives can be Riemann integrable, nowhere monotone, and without local extrema, expanding understanding of differentiability and integrability.

Findings

Constructed a nowhere monotone Riemann integrable derivative without local extrema.

Built a differentiable function with Riemann integrable derivative having isolated local extrema not being inflection points.

Provided examples challenging traditional views on derivatives and extrema.

Abstract

We construct a nowhere monotone Riemann integrable derivative which has no local extrema. We also construct a differentiable function $G$ such that $G'$ is Riemann integrable and has an isolated local extrema which is not an inflection point of $G$.