  The objective of this paper is to deepen the understanding of the connection
between the continuous and smooth dependence of solutions on initial conditions
and the regularity of the history functionals for retarded functional
differential equations. We consider some differential equation with a single
constant delay with the history space of $L^p$-type and obtain the above
dependence result by assuming the growth rate of the nonlinearity and its
derivative. The corresponding history functional is discontinuous, and it
becomes clear that there are the continuity and the smoothness of the
composition operators (also called the superposition operators or the Nemytskij
operators) between $L^p$-spaces behind the dependence results.
