Define f : [0, 1] → ℑ by f(x) = {0 if x is irrational, 1/n if x is rational and x = m/n in lowest terms with m and n in ℕ, 1 if x = 0. From Example 4.8 and Exercise 8 in Section 4.2, f is continuous at every irrational number and discontinuous at every rational number. Show that f is in ℓ[0, 1] and find ∫[0,1] f. [Hint: Since each lower sum is 0, given ε > 0, we want a P such that U(P, f) < ε. In [0, 1] there are only a finite number (say r) of points m/n with 1/n > ε/2. Choose P with ||P|| < ε/4r.]