Let (x) := x2|cos(p/x)| for x [0, 1] and (0) := 0. Then is continuous

Let Ψ(x) := x2|cos(p/x)| for x ∈ [0, 1] and Ψ(0) := 0. Then Ψ is continuous on [0, 1] and Ψʹ(x) exists for x ∉ E1 := {ak}. Let Ψ(x) := Ψʹ(x) for x ∉ E1, and ψ(x) := 0 for x ∈ E1. Show that ψ is bounded on [0, 1] and (using Exercise 7.2.11) that ψ ∈ R[0, 1]. Show that ∫ac ψ = Ψ (b) - Ψ(a) for a, b ∈ [0, 1]. Also show that |c| ∈ R[0, 1].