Let (x) := x|cos(/x)| for x [0, 1] and let (0+ := 0. Then is

Let Φ(x) := x|cos(π/x)| for x ∈ [0, 1] and let Φ(0+ := 0. Then Φ is continuous on [0, 1] and Φʹ(x) exists for x ∉ E1 := {0} ⋃ {ak : k ∈ N}, where ak := 2/(2k + 1). Let ω(x) := Φʹ(x) for x ∉ E and ω(x) := 0 for x ∈ E. Show that w is not bounded on [0,1]. Using the Fundamental Theorem 10.1.9 with E countable, conclude that ω ∈ R*[0, 1] and that ∫ba ω = Φ(b) - Φ(a) for a, b ∈ [0, 1]. As in Exercise 19, show that |ω| ∉ R*[0, 1].