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16 1. Preliminary Notions Problem 1.7. Let f : [a, b] → R be a function continuous at the point c ∈ (a, b). Consider the function g : [a, b] → R defined by g(x) = 0 if a ≤ x ≤ c, 1 if c x ≤ b. Prove that b a f dg = f(c). Problem 1.8. Let a c1 c2 c3 b and let f : [a, b] → R be a function which is continuous at the points ci (i = 1, 2, 3). Moreover, let β1,β2,β3,β4 ∈ R and let g : [a, b] → R be defined by g(x) = ⎧ ⎪β ⎪ ⎨ ⎪β3 ⎪ ⎩ 1 if a ≤ x ≤ c1, β2 if c1 x ≤ c2, if c2 x ≤ c3, β4 if c3 x ≤ b. Show that b a f dg = (β2 − β1)f(c1) + (β3 − β2)f(c2) + (β4 − β3)f(c3). Problem 1.9. (a) Using definition ( 1.7) or ( 1.8), prove that the following functions belong to the set of slowly oscillating functions L: log x, log3 x, e √ log x . (b) Prove that the following functions do not belong to L: 1 x , sin x. (c) Prove that the following functions are regularly varying: x2 log x, x log x , x1+1/x. Problem 1.10. Consider the function f : N → {0, 1} defined by f(1) = 1 and, for each integer n ≥ 2, by f(n) = 1 if 22m n ≤ 22m+1, 0 if 22m+1 n ≤ 22m+2. Show that the set A = {n ∈ N : f(n) = 1} does not have a density. Problem 1.11. Let A ⊂ N be a set of zero density and let a1 be its smallest element. Also, let L : [a1, +∞) → R+ be an increasing function which is continuous and differentiable on [a1, +∞). Assume moreover that L (x) = O(1). Prove that the following two statements are equivalent: