Summary: By using averaging functions, new interval oscillation criteria are established for the second-order functional-differential equation
that are different from most known ones in the sense that they are based on information only on a sequence of subintervals of , rather than on the whole half-line. Our results can be applied to three cases: ordinary, delay, and advance differential equations. In the case of half-linear functional-differential equations, our criteria implies that the delay and advance cases do not affect the oscillation. In particular, several examples are given to illustrate the importance of our results.