In mathematics, Darboux-Froda's theorem, named after Alexandru Froda, a Romanian mathematician, describes the set of discontinuities of a monotone real-valued function of a real variable. Usually, this theorem appears in literature without a name. It was written in A. Froda' thesis in 1929 .. As it is acknowledged in the thesis, it is in fact due Jean Gaston Darboux
- Consider a function f of real variable x with real values defined in a neighborhood of a point
x
0
and the function f is discontinuous at the point on the real axis
x
=
x
0
. We will call a removable discontinuity or a jump discontinuity a discontinuity of the first kind.
- Denote
f
(
x
+
0
)
:=
lim
h
↘
0
f
(
x
+
h
)
and
f
(
x
−
0
)
:=
lim
h
↘
0
f
(
x
−
h
)
. Then if
f
(
x
0
+
0
)
and
f
(
x
0
−
0
)
are finite we will call the difference
f
(
x
0
+
0
)
−
f
(
x
0
−
0
)
the jump of f at
x
0
.
If the function is continuous at
x
0
then the jump at
x
0
is zero. Moreover, if
f
is not continuous at
x
0
, the jump can be zero at
x
0
if
f
(
x
0
+
0
)
=
f
(
x
0
−
0
)
≠
f
(
x
0
)
.
Let f be a real-valued monotone function defined on an interval I. Then the set of discontinuities of the first kind is at most countable.
One can prove that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind. With this remark Froda's theorem takes the stronger form:
Let f be a monotone function defined on an interval
I
. Then the set of discontinuities is at most countable.
Let
I
:=
[
a
,
b
]
be an interval and
f
defined on
I
an increasing function. We have
f
(
a
)
≤
f
(
a
+
0
)
≤
f
(
x
−
0
)
≤
f
(
x
+
0
)
≤
f
(
b
−
0
)
≤
f
(
b
)
for any
a
<
x
<
b
. Let
α
>
0
and let
x
1
<
x
2
<
⋯
<
x
n
be
n
points inside
I
at which the jump of
f
is greater or equal to
α
:
f
(
x
i
+
0
)
−
f
(
x
i
−
0
)
≥
α
,
i
=
1
,
2
,
…
,
n
We have
f
(
x
i
+
0
)
≤
f
(
x
i
+
1
−
0
)
or
f
(
x
i
+
1
−
0
)
−
f
(
x
i
+
0
)
≥
0
,
i
=
1
,
2
,
…
,
n
. Then
f
(
b
)
−
f
(
a
)
≥
f
(
x
n
+
0
)
−
f
(
x
1
−
0
)
=
∑
i
=
1
n
[
f
(
x
i
+
0
)
−
f
(
x
i
−
0
)
]
+
+
∑
i
=
1
n
−
1
[
f
(
x
i
+
1
−
0
)
−
f
(
x
i
+
0
)
]
≥
∑
i
=
1
n
[
f
(
x
i
+
0
)
−
f
(
x
i
−
0
)
]
≥
n
α
and hence:
n
≤
f
(
b
)
−
f
(
a
)
α
.
Since
f
(
b
)
−
f
(
a
)
<
∞
we have that the number of points at which the jump is greater than
α
is finite or zero.
We define the following sets:
S
1
:=
{
x
:
x
∈
I
,
f
(
x
+
0
)
−
f
(
x
−
0
)
≥
1
}
,
S
n
:=
{
x
:
x
∈
I
,
1
n
≤
f
(
x
+
0
)
−
f
(
x
−
0
)
<
1
n
−
1
}
,
n
≥
2.
We have that each set
S
n
is finite or the empty set. The union
S
=
∪
n
=
1
∞
S
n
contains all points at which the jump is positive and hence contains all points of discontinuity. Since every
S
i
,
i
=
1
,
2
,
…
is at most countable, we have that
S
is at most countable.
If
f
is decreasing the proof is similar.
If the interval
I
is not closed and bounded (and hence by Heine–Borel theorem not compact) then the interval can be written as a countable union of closed and bounded intervals
I
n
with the property that any two consecutive intervals have an endpoint in common:
I
=
∪
n
=
1
∞
I
n
.
If
I
=
(
a
,
b
]
,
a
≥
−
∞
then
I
1
=
[
α
1
,
b
]
,
I
2
=
[
α
2
,
α
1
]
,
…
,
I
n
=
[
α
n
,
α
n
−
1
]
,
…
where
{
α
n
}
n
is a strictly decreasing sequence such that
α
n
→
a
.
In a similar way if
I
=
[
a
,
b
)
,
b
≤
+
∞
or if
I
=
(
a
,
b
)
−
∞
≤
a
<
b
≤
∞
.
In any interval
I
n
we have at most countable many points of discontinuity, and since a countable union of at most countable sets is at most countable, it follows that the set of all discontinuities is at most countable.