1. The problem statement, all variables and given/known data Let f(x) be deﬁned on [0,1] by f(x) = 1 if x is rational f(x) = 0 if x is irrational. Is f integrable on [0,1]? You may use the fact that between any two rational numbers there exists an irrational number, and between any two irrational numbers there exists a rational number. 2. Relevant equations 3. The attempt at a solution Divide into n sub-intervals. Δxi=1/n U(f,Pn) = Ʃ(f(Ui)Δxi) = [itex]\sum(1)(1/n)[/itex] = 1/n L(f,Pn) = Ʃ[f(li)](Δxi) = [itex]\sum(0)(1/n)[/itex] = 0 As n[itex]\rightarrow[/itex] [itex]\infty[/itex] both U(f,Pn) and L(f,Pn) [itex]\rightarrow[/itex] 0 Therefore [itex]\int f(x)dx[/itex] = 0 Is this correct?