Darboux like functions

Abstract

A function \(f:\mathbb{R}\to\mathbb{R}\) is said to have the intermediate value property provided that if \(p\) and \(q\) are real numbers such that \(p\neq q\) and \(f(p)\lt f(q)\), then for every \(y\in (f(p),f(q))\) there exists a number \(x\) between \(p\) and \(q\) with \(f(x)=y\). In 1875, G. Darboux showed that there exist functions with the intermediate value property that are not continuous \cite{22}. Because of his work with functions having the intermediate value property, these functions are called Darboux functions.