Determination of functions by metric slopes

TL;DR

This paper proves that in metric spaces, continuous functions with certain properties are uniquely determined by their metric slopes and critical values, highlighting a fundamental relationship between a function's shape and its slope behavior.

Contribution

It establishes a novel uniqueness result linking functions to their metric slopes and critical values in metric spaces.

Findings

Functions with compact sublevel sets are uniquely determined by their metric slopes.

Finite metric slopes combined with critical values suffice for function determination.

The result extends the understanding of function characterization in metric spaces.

Abstract

We show that in a metric space, any continuous function with compact sublevel sets and finite metric slope is uniquely determined by the slope and its critical values.