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:
