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) =

⎧

⎪

⎪

⎪β1

⎪

⎨β2

⎪

⎪

⎪β3

⎪

⎩β4

if a ≤ x ≤ c1,

if c1 x ≤ c2,

if c2 x ≤ c3,

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: