This paper gives examples of two quasicontinuous functions whose second iterates are discontinuous everywhere. It is well-known that every quasicontinuous function has a dense---indeed, residual---set of points of continuity; our counter-examples show that this property does not hold for compositions of such functions.
"Quasicontinuous Functions with Everywhere Discontinuous Iterates." Real Anal. Exchange 33 (1) 161 - 166, 2007/2008.