The author considers the functional differential equation (1) \(\dot x(t)=\int^ t_ a[d_ sQ(t,s)]f(t,x(s))\), \(t\in[t_ 0,T)\), where \(- \infty<c\leq t_ 0<T<\infty\) and \((H_ 1)\) \(f:[t_ 0,T)\times R^ n\to R^ n\) is a continuous function satisfying the conditions: \(| f(t,x_ 1)-f(t,x_ 2)|\leq L(t)| x_ 1-x_ 2|\), \(| f(t,0)|\leq CL(t)\), where \(L(t):\) \([t_ 0,T)\to[0,\infty)\) is continuous and \(C\) is a constant; \((H_ 2)\) \(Q:[t_ 0,T)\times[c,T)\to M_ n\) is a matrix-function of locally bounded variation in the second variable for every fixed value of the first variable and \(t\to\int^ t_ c[d_ sQ(t,s)]y(t,s)\) is continuous on \([t_ 0,T)\) for every continuous function \(y(t,s)\) on \(\{(t,s):t\in[t_ 0,T),s\in[c,t]\}\). Sufficient conditions are stated for \(\varphi\)-completeness at \(T\) and pointwise \(\varphi\)-completeness of the equation (1).