The Fixed Point of the Composition of Derivatives

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.