The main topics of this study are convergence and topologization. Integration on a compact interval on the real line is treated with Riemannian sums for various integration bases. General results are specified to a spectrum of integrations, including Lebesgue integration, the Denjoy integration in the restricted sense, the integrations introduced by Pfeffer and by Bongiorno, and many others. Moreover, some relations between integration and differentiation are made clear. The volume is self-contained, and should be of interest to specialists in the field of real functions, and to students, since only the basics of mathematical analysis and vector spaces are required.