Problem 1: Let f be the Dirichlet function over [0, 1] defined by f(x) =0, if x is rational; f(x) =1, if x is irrational. (i) For any partition I' of the interval [0, 1], find the lower and upper Riemann sums (also known as the Darboux sums) Lr and Ur of f. (ii) Show that f is not Riemannian integrable. (iii) Show that f is Lebesgue integrable and find its Lebesgue integral. (iv) Show that f is nowhere continuous in [0, 1]