# Recognizing difference quotients of real functions

## TL;DR

This paper explores methods to identify when a two-variable function is a difference quotient of a real function and provides techniques to reconstruct the original function from it.

## Contribution

It introduces three independent methods to recognize difference quotients and recover the original functions, advancing understanding of their characterization.

## Key findings

- Developed three methods for recognizing difference quotients.
- Provided techniques for reconstructing original functions from difference quotients.
- Enhanced understanding of the structure of difference quotients.

## Abstract

For a real function $f:[0,1]\to\mathbb{R}$, the difference quotient of $f$ is the function of two real variables $\operatorname{DQ}_f(a,b)=\dfrac{f(b)-f(a)}{b-a}$, which we view as defined on the triangle $\mathcal{T}=\{(a,b):0\leq a<b\leq1\}$. In this paper we investigate how to determine whether a given function of two variables $H(a,b)$ is the difference quotient of some real function $f(x)$. We develop three independent methods for recognizing such a function $H$ as a difference quotient, and corresponding methods for recovering the underlying function $f$ from $H$.