In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, Froda's theorem, named after Alexandru Froda

Alexandru Froda

Alexandru Froda was a well-known Romanian mathematician with important contributions in the field of mathematical analysis, algebra, number theory and rational mechanics. In his 1929 thesis he proved what is now known as Froda's theorem.-Life:Alexandru Froda was born in Bucharest in 1894...

, describes the set of discontinuities of a (monotone) real-valued function

Real-valued function

In mathematics, a real-valued function is a function that associates to every element of the domain a real number in the image....

 of a real variable. Usually, this theorem appears in literature without Froda's name being mentioned. However, this result was first proven by A. Froda in 1929.

Definitions

1. Consider a function of real variable with real values defined in a neighborhood of a point and the function is discontinuous at the point on the real axis . We will call a removable discontinuity or a jump discontinuity a discontinuity of the first kind.
2. Denote and . Then if and are finite we will call the difference the jump of f at .

It is obvious that if the function is continuous at then the jump at is zero. Moreover, if is not continuous in , the jump can be zero at if .

Precise statement

Let f be a monotone function defined on an interval I. Then the set of discontinuities of the first kind is at most countable.

Proof

Let be an interval and defined on an increasing

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

 function. We have


for any . Let and let be points inside at which the jump of is greater or equal to :


We have or .
Then


and hence: .

Since we have that the number of points at which the jump is greater than is finite or zero.

We define the following sets:
,


We have that each set is finite or the empty set

Empty set

In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...

. The union
contains all points at which the jump is positive and hence contains all points of discontinuity. Since every is at most countable, we have that is at most countable.

If is decreasing

Monotonic function

In mathematics, a monotonic function is a function that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory....

 the proof is similar.

If the interval is not closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 and bounded

Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...

 (and hence by Heine-Borel theorem not compact) then the interval can be written as a countable union of closed and bounded intervals with the property that any two consecutive intervals have an endpoint in common:

If then where is a strictly decreasing sequence

Sequence

In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

 such that In a similar way if or if .

In any interval 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.

Remark

One can prove that all points of discontinuity of a monotone real-valued function defined on an interval are either removable or 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 . Then the set of discontinuities is at most countable.

See also

• Continuous function
  Continuous function
  In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
  
  
• Classification of discontinuities
  Classification of discontinuities
  Continuous functions are of utmost importance in mathematics 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...
  
  

Let f be a monotone function defined on an interval I. Then the set of discontinuities of f is at most countable.