The bending problem of Euler-Bernoulli discontinuous beams is a classic topic in mechanics. In this paper stepped beams with internal springs are addressed based on the theory of generalized functions. It is shown that in this context a closed-form expression may be given to the Green's functions due to point forces and, based on these, to the beam response to arbitrary loads, for any set of boundary conditions. The proposed solution method may be presented in a regular course in Mechanics of Solids and Strength of Materials for undergraduate students. It does not require an advanced knowledge of the theory of generalized functions but the knowledge of only a few basic concepts, most of which are generally presented in other courses such as, for instance, Dynamics of Structures. It is hoped that it may help students to address in a simple and effective way the many engineering applications involving discontinuous beams.