Consider the periodic boundary value problem \((*)\) \(dx/dt = f(t,x)\), \(x(0)=x(T)\) where \(f:[0,T] \times \mathbb{R} \rightarrow \mathbb{R}\) is a Carathéodory function. The authors introduce the concept of piecewise absolutely continuous lower and upper solutions to \((*)\) (which can have jumps) and prove that the existence of ordered piecewise absolutely continuous lower and upper solutions implies the existence of at least one solution to \((*)\). Two examples are considered. The authors establish a connection of the method under consideration to differential equations with impulses.