Since 1980’s differential equations with piecewise constant arguments have been at- tracted great deal of attention of researchers in mathematical and some of the others fields in science. Piecewise constant systems exist in a widely expanded areas such as biomedicine, chemistry, mechanical engineering, physics, etc. These kind of equations such as Eq.(1) are similar in structure to those found in certain sequential-continuous models of disease dynamics [1]. In 1994, Dai and Sing [2] studied the oscillatory motion of spring-mass systems with subject to piecewise constant forces of the form f ( x [ t ]) or f ([ t ]) . Later, they improved an analytical and numerical method for solving linear and nonlinear vibration problems and they showed that a function f ([ N ( t )] /N ) is a good approximation to the given continuous function f ( t ) if N is sufficiently large [3]. In 1984, Cooke and Wiener [4] studied oscillatory and periodic solutions of a linear differential equation with piecewise constant argument and they note that such equa- tions are comprehensively related to impulsive and difference equations. After this work, oscillatory and periodic solutions of linear differential equations with piecewise constant arguments have been dealt with by many authors [5, 6, 7] and the references cited therein. On the other hand, in 1994, the case of studying discontinuous solutions of differ- ential equations with piecewise continuous arguments has been proposed as an open problem by Wiener [8]. Due to this open problem, some impulsive differential equa- tions with piecewise constant arguments have been studied [9, 10, 11]. Moreover, the monographs [12, 13] includes many results on the theory of differential equations with piecewise constant arguments. Now, our aim is to consider the Wiener’s open problem for the equation (1)-(2). Moreover, as we know there is only one work on nonimpulsive delay differential equa- tions with continuous and piecewise constant arguments [14]. In this respect, we obtain sufficient conditions for the existence of oscillatory solutions of Eq. (1)-(2). 2