Sets where Lip $f$ is infinite and lip $f$ is finite

TL;DR

This paper characterizes subsets of real numbers where a continuous function has finite Lipschitz constant everywhere except on a specific set where it becomes infinite, providing a detailed description of such sets.

Contribution

It offers a complete characterization of subsets of real numbers where a continuous function's Lipschitz constant is infinite exactly on those sets, advancing understanding of Lipschitz behavior.

Findings

Identifies conditions for sets where lip f is infinite

Describes the structure of sets with finite lip f everywhere except on E

Provides a construction method for such functions

Abstract

We characterize the subsets $E \subset \mathbb{R}$ for which there exists a continuous real valued function $f: \mathbb{R}\to\mathbb{R}$ such that lip $f$ is finite everywhere and Lip $f$ is infinite exactly on $E$.