The paper contains a significant contribution to the classical method of upper and lower solutions for a nonlinear first-order differential equation \(x'(t)=f(t,x(t)), t\in I=[0,1]\), with \(f:I\times \mathbb{R}\to \mathbb{R}\). The author generalizes the previous results in several directions: 1. Function \(f\) is allowed to be discontinuous, satisfying certain conditions introduced by \textit{E. R. Hassan} and \textit{W. Rzymowski} [Nonlinear. Anal., Theory Methods Appl. 37A, No. 8, 997-1017 (1999; Zbl 0949.34005)], which are weaker than the so-called Carathéodory conditions. 2. Classical continuous upper and lower solutions are replaced by functions of bounded variation. 3. Discontinuous and nonlinear boundary conditions with functional dependence are considered. The main result establishes the existence of the minimal and the maximal solution to a general boundary value problem between a lower and an upper solution. Two examples are included which show that the generalization achieved allows to prove the existence of the extremal solutions to some problems not included in previous settings, and also to improve the approximation to such solutions, because the upper and lower solutions are taken from a wider class of functions.