I applied a linear regression (continuous response, two continuous predictors and one categorical variable). The plot of residuals and fitted values is something like two different clusters. Can I say that the assumptions are satisfied in this example?
If you see any pattern in a residuals vs fitted plot different from "the good one" (which is the case when values spread randomly in an horizontal band around 0), this means that "something remains to be explained" and that the assumptions of the linear model are not fully satisfied (cf. https://data.library.virginia.edu/diagnostic-plots/ or https://onlinecourses.science.psu.edu/stat501/node/277 for instance). In your particular case, I think you miss some supplementary binary variable in your model.