Abstract <p> We study a boundary value problem for a diffusion equation with piecewise constantarguments. By separation of variables, the analysis of this boundary value problem is reduced tothat of the existence conditions for solutions of initial value problems for a differential equationwith piecewise constant arguments. Conditions for the existence of infinitely many solutions or theemptiness of the solution set are established for the differential equation in question, and explicitformulas for these solutions are obtained. Several examples are given to illustrate the results.</p>