The Fixed Point of the Composition of Derivatives
  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. 
M\'arton Elekes, Tam\'as Keleti, Vilmos Prokaj
math.CA