  Integrating with respect to functions which are constant on intervals whose
bounds are discontinuity points (of those functions) is frequent in many
branches of Mathematics, specially in stochastic processes. For such functions
and alike extension, a comparison between Riemann-Stieltjes and
Lebesgue-Stieltjes integration and the integrals formulas leads to interesting
facts for students (as complements of Measure Theory and Integrations) and for
practitioners and and researchers. We undergone conditions of existence the
Riemann-Stieltjes integrals related to that type of function and compare the
results with what should be expected for Lebesgue-Stieltjes theory.
