13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Classification of discontinuities | 2/4 | https://en.wikipedia.org/wiki/Classification_of_discontinuities | reference | science, encyclopedia | 2026-05-05T09:08:17.602531+00:00 | kb-cron |
The set
D
{\displaystyle D}
is an
F
σ
{\displaystyle F_{\sigma }}
set. The set of points at which a function is continuous is always a
G
δ
{\displaystyle G_{\delta }}
set (see). If on the interval
I
,
{\displaystyle I,}
f
{\displaystyle f}
is monotone then
D
{\displaystyle D}
is at most countable and
D
=
J
.
{\displaystyle D=J.}
This is Froda's theorem. Tom Apostol follows partially the classification above by considering only removable and jump discontinuities. His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem. With the same purpose, Walter Rudin and Karl R. Stromberg study also removable and jump discontinuities by using different terminologies. However, furtherly, both authors state that
R
∪
J
{\displaystyle R\cup J}
is always a countable set (see). The term essential discontinuity has evidence of use in mathematical context as early as 1889. However, the earliest use of the term alongside a mathematical definition seems to have been given in the work by John Klippert. Therein, Klippert also classified essential discontinuities themselves by subdividing the set
E
{\displaystyle E}
into the three following sets:
E
1
=
{
x
0
∈
I
:
lim
x
→
x
0
−
f
(
x
)
and
lim
x
→
x
0
+
f
(
x
)
do not exist in
R
}
,
{\displaystyle E_{1}=\left\{x_{0}\in I:\lim _{x\to x_{0}^{-}}f(x){\text{ and }}\lim _{x\to x_{0}^{+}}f(x){\text{ do not exist in }}\mathbb {R} \right\},}
E
2
=
{
x
0
∈
I
:
lim
x
→
x
0
−
f
(
x
)
exists in
R
and
lim
x
→
x
0
+
f
(
x
)
does not exist in
R
}
,
{\displaystyle E_{2}=\left\{x_{0}\in I:\ \lim _{x\to x_{0}^{-}}f(x){\text{ exists in }}\mathbb {R} {\text{ and }}\lim _{x\to x_{0}^{+}}f(x){\text{ does not exist in }}\mathbb {R} \right\},}
E
3
=
{
x
0
∈
I
:
lim
x
→
x
0
−
f
(
x
)
does not exist in
R
and
lim
x
→
x
0
+
f
(
x
)
exists in
R
}
.
{\displaystyle E_{3}=\left\{x_{0}\in I:\ \lim _{x\to x_{0}^{-}}f(x){\text{ does not exist in }}\mathbb {R} {\text{ and }}\lim _{x\to x_{0}^{+}}f(x){\text{ exists in }}\mathbb {R} \right\}.}
Of course
E
=
E
1
∪
E
2
∪
E
3
.
{\displaystyle E=E_{1}\cup E_{2}\cup E_{3}.}
Whenever
x
0
∈
E
1
,
{\displaystyle x_{0}\in E_{1},}
x
0
{\displaystyle x_{0}}
is called an essential discontinuity of first kind. Any
x
0
∈
E
2
∪
E
3
{\displaystyle x_{0}\in E_{2}\cup E_{3}}
is said an essential discontinuity of second kind. Hence he enlarges the set
R
∪
J
{\displaystyle R\cup J}
without losing its characteristic of being countable, by stating the following:
The set
R
∪
J
∪
E
2
∪
E
3
{\displaystyle R\cup J\cup E_{2}\cup E_{3}}
is countable.
== Rewriting Lebesgue's theorem == When
I
=
[
a
,
b
]
{\displaystyle I=[a,b]}
and
f
{\displaystyle f}
is a bounded function, it is well-known of the importance of the set
D
{\displaystyle D}
in the regard of the Riemann integrability of
f
.
{\displaystyle f.}
In fact, Lebesgue's theorem (also named Lebesgue-Vitali) theorem) states that
f
{\displaystyle f}
is Riemann integrable on
I
=
[
a
,
b
]
{\displaystyle I=[a,b]}
if and only if
D
{\displaystyle D}
is a set with Lebesgue's measure zero. In this theorem seems that all type of discontinuities have the same weight on the obstruction that a bounded function
f
{\displaystyle f}
be Riemann integrable on
[
a
,
b
]
.
{\displaystyle [a,b].}
Since countable sets are sets of Lebesgue's measure zero and a countable union of sets with Lebesgue's measure zero is still a set of Lebesgue's measure zero, we are seeing now that this is not the case. In fact, the discontinuities in the set
R
∪
J
∪
E
2
∪
E
3
{\displaystyle R\cup J\cup E_{2}\cup E_{3}}
are absolutely neutral in the regard of the Riemann integrability of
f
.
{\displaystyle f.}
The main discontinuities for that purpose are the essential discontinuities of first kind and consequently the Lebesgue-Vitali theorem can be rewritten as follows:
A bounded function,
f
,
{\displaystyle f,}
is Riemann integrable on
[
a
,
b
]
{\displaystyle [a,b]}
if and only if the correspondent set
E
1
{\displaystyle E_{1}}
of all essential discontinuities of first kind of
f
{\displaystyle f}
has Lebesgue's measure zero. The case where
E
1
=
∅
{\displaystyle E_{1}=\varnothing }
correspond to the following well-known classical complementary situations of Riemann integrability of a bounded function
f
:
[
a
,
b
]
→
R
{\displaystyle f:[a,b]\to \mathbb {R} }
:
If
f
{\displaystyle f}
has right-hand limit at each point of
[
a
,
b
[
{\displaystyle [a,b[}
then
f
{\displaystyle f}
is Riemann integrable on
[
a
,
b
]
{\displaystyle [a,b]}
(see) If
f
{\displaystyle f}
has left-hand limit at each point of
]
a
,
b
]
{\displaystyle ]a,b]}
then
f
{\displaystyle f}
is Riemann integrable on
[
a
,
b
]
.
{\displaystyle [a,b].}
If
f
{\displaystyle f}
is a regulated function on
[
a
,
b
]
{\displaystyle [a,b]}
then
f
{\displaystyle f}
is Riemann integrable on
[
a
,
b
]
.
{\displaystyle [a,b].}
=== Examples === Thomae's function is discontinuous at every non-zero rational point, but continuous at every irrational point. One easily sees that those discontinuities are all removable. By the first paragraph, there does not exist a function that is continuous at every rational point, but discontinuous at every irrational point. The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere. These discontinuities are all essential of the first kind too. Consider now the ternary Cantor set
C
⊂
[
0
,
1
]
{\displaystyle {\mathcal {C}}\subset [0,1]}
and its indicator (or characteristic) function