The main aim of this paper is to develop some basic theories of solutions of neutral stochastic functional differential equations (NSFDEs). Firstly, by fixed point theorem, the local existence of solutions for NSFDEs is studied under only the continuity condition for drift coefficient and diffusion coefficient. Secondly, a uniqueness theorem is established by comparison methods. Furthermore, the continuation theorem is obtained under which the global existence of solutions is presented for NSFDEs.
