# Barrier functions in the subdifferential theory

## TL;DR

This paper introduces a novel approach using barrier functions to prove that monotonicity of subdifferentials implies convexity of functions, addressing technical challenges with lower semicontinuous functions.

## Contribution

The paper presents a new method employing barrier functions to establish the Correa-Jofré-Thibault theorem, enhancing the theoretical toolkit for subdifferential analysis.

## Key findings

- Barrier functions facilitate handling lower semicontinuous functions.
- The new proof simplifies existing technical difficulties.
- The method confirms the link between monotonicity and convexity.

## Abstract

We present a new method for proving Correa-Jofr\'e-Thibault theorem that monotonicity of subdifferential implies convexity of the function. This new method is based on barrier functions. Barrier functions help overcome some of the main technical difficulties when working with lower semicontinuous functions.