In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...
, a well-known theorem describes the set of
discontinuities of a
monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called ''real f ...
of a real variable; all discontinuities of such a (monotone) function are necessarily
jump discontinuities and there are at most
countably many
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
of them.
Usually, this theorem appears in literature without a name. It is called Froda's theorem in some recent works; in his 1929 dissertation,
Alexandru Froda stated that the result was previously well-known and had provided his own elementary proof for the sake of convenience. Prior work on discontinuities had already been discussed in the 1875 memoir of the French mathematician
Jean Gaston Darboux
Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician.
Life
According this birth certificate he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midnig ...
.
Definitions
Denote the
limit from the left by
and denote the
limit from the right by
If
and
exist and are finite then the difference
is called the jump of
at
Consider a real-valued function
of real variable
defined in a neighborhood of a point
If
is discontinuous at the point
then the discontinuity will be a
removable discontinuity, or an
essential discontinuity
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of a ...
, or a
jump discontinuity
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of a ...
(also called a discontinuity of the first kind).
If the function is continuous at
then the jump at
is zero. Moreover, if
is not continuous at
the jump can be zero at
if
Precise statement
Let
be a real-valued
monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
function defined on an
interval 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 the theorem takes the stronger form:
Let
be a monotone function defined on an interval
Then the set of discontinuities is at most countable.
Proofs
This proof starts by proving the special case where the function's domain is a closed and bounded interval
The proof of the general case follows from this special case.
Proof when the domain is closed and bounded
Two proofs of this special case are given.
Proof 1
Let