Discontinuities of monotone functions

In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a function are necessarily jump discontinuities and there are at most countably many 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.

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, or a jump discontinuity.
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 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 be an interval and let be a non-decreasing function.
Then for any
Let and let be points inside at which the jump of is greater or equal to :
For any so that
Consequently,
and hence
Since we have that the number of points at which the jump is greater than is finite.
Define the following sets:
Each set is finite or the empty set. The union
contains all points at which the jump is positive and hence contains all points of discontinuity. Since every is at most countable, their union is also at most countable.
If is non-increasing then the proof is similar. This completes the proof of the special case where the function's domain is a closed and bounded interval.

Proof 2

For a monotone function, let mean that is monotonically non-decreasing and let mean that is monotonically non-increasing. Let is a monotone function and let denote the set of all points in the domain of at which is discontinuous.
Because has a jump discontinuity at so there exists some rational number that lies strictly in between .
It will now be shown that if are distinct, say with then
If then implies so that
If on the other hand then implies so that
Either way,
Thus every is associated with a unique rational number.
Since is countable, the same must be true of

Proof of general case

Suppose that the domain of is equal to a union of countably many closed and bounded intervals; say its domain is .
It follows from the special case proved above that for every index the restriction of to the interval has at most countably many discontinuities; denote this set of discontinuities by
If has a discontinuity at a point in its domain then either is equal to an endpoint of one of these intervals or else there exists some index such that in which case must be a point of discontinuity for .
Thus the set of all points of at which is discontinuous is a subset of which is a countable set so that its subset must also be countable.
In particular, because every interval of real numbers can be written as a countable union of closed and bounded intervals, it follows that any monotone real-valued function defined on an interval has at most countable many discontinuities.
To make this argument more concrete, suppose that the domain of is an interval that is not closed and bounded.
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 such that In a similar way if or if
In any interval there are 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.

Jump functions

Examples. Let ₁ < ₂ < ₃ < ⋅⋅⋅ be a countable subset of the compact interval and let μ₁, μ₂, μ₃,... be a positive sequence with finite sum. Set
where χ_A denotes the characteristic function of a compact interval. Then is a non-decreasing function on, which is continuous except for jump discontinuities at for ≥ 1. In the case of finitely many jump discontinuities, is a step function. The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions.
More generally, the analysis of monotone functions has been studied by many mathematicians, starting from Abel, Jordan and Darboux. Following, replacing a function by its negative if necessary, only the case of non-negative non-decreasing functions has to be considered. The domain can be finite or have ∞ or −∞ as endpoints.
The main task is to construct monotone functions — generalising step functions — with discontinuities at a given denumerable set of points and with prescribed left and right discontinuities at each of these points.
Let lie in and take λ₁, λ₂, λ₃,... and μ₁, μ₂, μ₃,... non-negative with finite sum and with λ + μ > 0 for each. Define
Then the jump function, or saltus-function, defined by
is non-decreasing on and is continuous except for jump discontinuities at for ≥ 1.
To prove this, note that sup || = λ + μ, so that Σ converges uniformly to. Passing to the limit, it follows that
if is not one of the 's.
Conversely, by a differentiation theorem of Lebesgue, the jump function is uniquely determined by the properties: being non-decreasing and non-positive; having given jump data at its points of discontinuity ; satisfying the boundary condition = 0; and having zero derivative almost everywhere.
Property can be checked following, and. Without loss of generality, it can be assumed that is a non-negative jump function defined on the compact, with discontinuities only in.
Note that an open set of is canonically the disjoint union of at most countably many open intervals ; that allows the total length to be computed ℓ= Σ ℓ. Recall that a null set is a subset such that, for any arbitrarily small ε' > 0, there is an open containing with ℓ < ε'. A crucial property of length is that, if and are open in, then ℓ + ℓ = ℓ + ℓ. It implies immediately that the union of two null sets is null; and that a finite or countable set is null.
Proposition 1. For > 0 and a normalised non-negative jump function, let be the set of points such that
for some, with < <. Then
is open and has total length ℓ) ≤ 4 ⁻¹ – ).
Note that consists the points where the slope of is greater that near. By definition is an open subset of, so can be written as a disjoint union of at most countably many open intervals =. Let be an interval with closure in and ℓ = ℓ/2. By compactness, there are finitely many open intervals of the form covering the closure of . On the other hand, it is elementary that, if three fixed bounded open intervals have a common point of intersection, then their union contains one of the three intervals: indeed just take the supremum and infimum points to identify the endpoints. As a result, the finite cover can be taken as adjacent open intervals,,... only intersecting at consecutive intervals. Hence
Finally sum both sides over.
Proposition 2. If is a jump function, then ' = 0 almost everywhere.
To prove this, define
a variant of the Dini derivative of. It will suffice to prove that for any fixed > 0, the Dini derivative satisfies ≤ almost everywhere, i.e. on a null set.
Choose ε > 0, arbitrarily small. Starting from the definition of the jump function = Σ , write = + with = Σ_≤ and = Σ_> where ≥ 1. Thus is a step function having only finitely many discontinuities at for ≤ and is a non-negative jump function. It follows that = ' + = except at the points of discontinuity of. Choosing sufficiently large so that Σ_> λ + μ < ε, it follows that is a jump function such that − < ε and ≤ off an open set with length less than 4ε/.
By construction ≤ off an open set with length less than 4ε/. Now set ε' = 4ε/ — then ε' and are arbitrarily small and ≤ off an open set of length less than ε'. Thus ≤ almost everywhere. Since could be taken arbitrarily small, and hence also ' must vanish almost everywhere.
As explained in, every non-decreasing non-negative function can be decomposed uniquely as a sum of a jump function and a continuous monotone function : the jump function is constructed by using the jump data of the original monotone function and it is easy to check that = − is continuous and monotone.