  In this paper, we establish a theory of well-posedness for delay differential
equations (DDEs) via notions of \textit{prolongations} and
\textit{$C^1$-prolongations}, which are continuous and continuously
differentiable extensions of histories to the right, respectively. In this
sense, this paper serves as a continuation and an extension of the previous
paper by this author (\cite{Nishiguchi 2017}). The results in \cite{Nishiguchi
2017} are applicable to various DDEs, however, the results in \cite{Nishiguchi
2017} cannot be applied to general class of state-dependent DDEs, and its
extendability is missing. We find this missing link by introducing notions of
($C^1$-) prolongabilities, regulation of topology by ($C^1$-) prolongations,
and Lipschitz conditions about ($C^1$-) prolongations, etc. One of the main
result claims that the continuity of the semiflow with a parameter generated by
the trivial DDEs $\dot{x} = v$ plays an important role for the well-posedness.
The results are applied to general class of state-dependent DDEs.
