Finite, countable and uncountable sets

Consider ¯x ∈ {0, 1} N defined by ¯x =(¯x 1 , ¯x 2 , ...), ¯x i =
¯x n 6= x nn . It follows that A is not countable.
½ 1 if xii =0
0 if x ii =1 .Then ¯x 6= x n ∀n ∈ N since
Remark 19 There is a bijection between P ({a 1 , ..., a n }) and {0, 1} N .
between P(N) and {0, 1} N .ThenP(N) is not countable.
In addition, there is a bijection
Theorem 20 If f : R → R is a nondecreasing function, then the set of points where f is discontinuous is
at most countable.
Proof. Consider the family {x λ } λ∈Λ
, the set of points where f is discontinuous. We need to show that Λ is
countable. Note that if x is discontinuous at x λ ,then
lim f(x) 6= lim f(x)
x→x − λ
x→x + λ
Since f is nondecreasing,
lim f(x) < lim f(x)
x→x − λ
x→x + λ
It is possible to find Q λ ∈ Q such that
lim f(x) x λ2 or
x λ1 x λ2 ⇒ Q λ1 >Q λ2 ⇒ g(λ 1 ) >g(λ 2 ). In the second case, x λ1