Continuity of Darboux Functions with Nice Finite Iterations

Abstract

A function that maps intervals into intervals is called a Darboux function. We prove that if $g$ is a continuous function that is non-constant on every non-empty open interval, and $f$ is a Darboux function such that, for every real number $x,$ $f^{n_{x}}(x)=g(x)$ for some positive integer $n_{x}$, and the set of all such $n_{x}$ is bounded, then $f$ is continuous. In the above statement, the hypothesis ``the set of all such $ n_{x} $ is bounded'' cannot be dropped. We also show that if $g$ is a continuous function that takes a constant value $k$ on some non-empty open interval $I$ and $k\in I$, then there exists a discontinuous Darboux function $f\:mathbb{R}\rightarrow \mathbb{R}$ with the property that, for every real number $x,$ $f^{n_{x}}(x)=g(x)$ for some positive integer $n_{x}\leq 2$. In the previous statement, if $k\notin I$, then no conclusion can be drawn about the function $f$.