Statements of the general results about existence and existence and uniqueness of local solutions for an abstract DDE x'(t)=f(t,x_t); proof of the existence and uniqueness theorem only in the case of f(t,\phi) continuous and Lipschitzian relative to \phi in each bounded set of $\R\times C([-r,0],\R^n), by using the contraction principle (rather than Schauder's fixed point theorem).

Consequences: continuous dependence of solutions relative to initial conditions; continuation of solutions (for the future).