WWe show that the existence of a loop of covering intervals for connected $G_\delta$ real functions implies the existence of a periodic point with its period equal to the length of the cycle. As a corollary we get that the composition of $N$ connected $G_\delta$ real functions from a compact interval into itself has a fixed point for every natural number $N$. In particular, the composition of finitely many derivatives from $[0,1]$ to $[0,1]$ has a fixed point. This solve the a problem from .
"Loops of intervals and Darboux Baire 1 fixed point problem.." Real Anal. Exchange 29 (1) 205 - 209, 2003-2004.