Thedelaydifferentialequationwithpiecewiseconstantargumentx′(t)+a(t)x(t)+b(t)x([t-k])=0isconsidered,wherea(t)andb(t)arecontinuousfunctionson[—k,∞),b(t)≥0,kisapositiveintegerand[·]denotesthegreatestintegerfunction.Somenewoscillationandnonoscillationconditionsareobtained.