The Fixed Point of the Composition of Derivatives

TL;DR

This paper proves that the composition of two derivatives or Darboux Baire-1 functions from [0,1] to [0,1] always has a fixed point, answering a longstanding open question.

Contribution

It establishes that the composition of two derivatives always has a fixed point, solving a problem posed by K. Ciesielski.

Findings

Composition of two derivatives on [0,1] always has a fixed point

Composition of two Darboux Baire-1 functions on [0,1] always has a fixed point

Uses Maximoff's Theorem to extend the fixed point result

Abstract

We give an affirmative answer to a question of K. Ciesielski by showing that the composition $f\circ g$ of two derivatives $f,g:[0,1]\to[0,1]$ always has a fixed point. Using Maximoff's Theorem we obtain that the composition of two $[0,1]\to[0,1]$ Darboux Baire-1 functions must also have a fixed point.