Recognizing difference quotients of real functions

Abstract

For a real function $f:[0,1]\to \mathbb{R}$, the difference quotient of $f$ is the function of two real variables ${\mathrm{DQ}}_f(a,b)=\frac{f(b)-f(a)}{b-a}$, which we view as defined on the triangle $\mathcal{T}=\{(a,b):0\le a<b\le 1\}$. 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 $H$.