Discontinuities of monotone functions explained

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 (monotone) 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.^[1] Prior work on discontinuities had already been discussed in the 1875 memoir of the French mathematician Jean Gaston Darboux.^[2]

Definitions

Denote the limit from the left by$$f\backslash left(x^-\backslash right)\; :=\; \backslash lim\_\; f(z)\; =\; \backslash lim\_\; f(x-h)$$and denote the limit from the right by$$f\backslash left(x^+\backslash right)\; :=\; \backslash lim\_\; f(z)\; =\; \backslash lim\_\; f(x+h).$$

If

f\left(x ^{+\right)} and f\left(x ^{-\right)} exist and are finite then the difference f\left(x ^{+\right)} - f\left(x ^{-\right)} is called the jump of f at x.

Consider a real-valued function

f of real variable x defined in a neighborhood of a point x. If f is discontinuous at the point x then the discontinuity will be a removable discontinuity, or an essential discontinuity, or a jump discontinuity (also called a discontinuity of the first kind).If the function is continuous at x then the jump at x is zero. Moreover, if f is not continuous at x, the jump can be zero at x if f\left(x ^{+\right)} = f\left(x ^{-\right)} ≠ f(x).

Precise statement

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 the 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.

Proofs

This proof starts by proving the special case where the function's domain is a closed and bounded interval

[a, b]. 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

I : = [a, b] be an interval and let f : I \to \R be a non-decreasing function (such as an increasing function).Then for any a < x < b, $$f(a)\; ~\backslash leq~\; f\backslash left(a^+\backslash right)\; ~\backslash leq~\; f\backslash left(x^-\backslash right)\; ~\backslash leq~\; f\backslash left(x^+\backslash right)\; ~\backslash leq~\; f\backslash left(b^-\backslash right)\; ~\backslash leq~\; f(b).$$Let \alpha > 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 \alpha :$$f\backslash left(x\_i^+\backslash right)\; -\; f\backslash left(x\_i^-\backslash right)\; \backslash geq\; \backslash alpha,\backslash \; i=1,2,\backslash ldots,n$$

For any

i =1 , 2 ,\ldots,n, f\left(x ^{+ \right)} _{i} \leq f\left(x ^{- \right)} _{i +1} so that f\left(x ^{- \right)} _{i +1} - f\left(x ^{+ \right)} _{i} \geq 0 . Consequently,and hence n \leq (f(b) - f(a))/(\alpha) .

Since

f(b) - f(a) < inf ty we have that the number of points at which the jump is greater than \alpha is finite (possibly even zero).

Define the following sets:$$S\_1:\; =\; \backslash left\backslash ,$$$$S\_n:\; =\; \backslash left\backslash ,\backslash \; n\backslash geq\; 2.$$

Each set

S _{n} is finite or the empty set. The unionS = c up ^{inf ty} _{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,\ldots} is at most countable, their union S is also at most countable.

If

f is non-increasing (or decreasing) then the proof is similar. This completes the proof of the special case where the function's domain is a closed and bounded interval.\blacksquare

Proof 2

So let

f : [a, b] \to \R is a monotone function and let D denote the set of all points d \in [a, b] in the domain of f at which f is discontinuous (which is necessarily a jump discontinuity).

Because

f has a jump discontinuity at d \in D, f\left(d ^{-\right)} ≠ f\left(d ^{+\right)} so there exists some rational number y _{d} \in \Q that lies strictly in between f\left(d ^{-\right)} and f\left(d ^{+\right)} (specifically, if f \nearrow then pick y _{d} \in \Q so that f\left(d ^{-\right)} < y _{d} < f\left(d ^{+\right)} while if f \searrow then pick y _{d} \in \Q so that f\left(d ^{-\right)} > y _{d} > f\left(d ^{+\right)} holds).

It will now be shown that if

d, e \in D are distinct, say with d < e, then y _{d} ≠ y _{e.} If f \nearrow then d < e implies f\left(d ^{+\right)} \leq f\left(e ^{-\right)} so that y _{d} < f\left(d ^{+\right)} \leq f\left(e ^{-\right)} < y _{e.} If on the other hand f \searrow then d < e implies f\left(d ^{+\right)} \geq f\left(e ^{-\right)} so that y _{d} > f\left(d ^{+\right)} \geq f\left(e ^{-\right)} > y _{e.} Either way, y _{d} ≠ y _{e.}

Thus every

d \in D is associated with a unique rational number (said differently, the map D \to \Q defined by d \mapsto y _{d} is injective).Since \Q is countable, the same must be true of D. \blacksquare

Proof of general case

Suppose that the domain of

f (a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is c up _{n} \left[a _{n,} b _{n\right]} (no requirements are placed on these closed and bounded intervals).It follows from the special case proved above that for every index n, the restriction f \ vert _{\left[a _{n,} b _{n\right]}} : \left[a _{n,} b _{n\right]} \to \R of f to the interval \left[a _{n,} b _{n\right]} has at most countably many discontinuities; denote this (countable) set of discontinuities by D _{n.} If f has a discontinuity at a point x _{0} \in c up _{n} \left[a _{n,} b _{n\right]} in its domain then either x _{0} is equal to an endpoint of one of these intervals (that is, x _{0} \in \left\{a _{1,} b _{1,} a _{2,} b _{2,} \ldots\right\} ) or else there exists some index n such that a _{n} < x _{0} < b _{n,} in which case x _{0} must be a point of discontinuity for f \ vert _{\left[a _{n,} b _{n\right]}} (that is, x _{0} \in D _{n} ).Thus the set D of all points of at which f is discontinuous is a subset of \left\{a _{1,} b _{1,} a _{2,} b _{2,} \ldots\right\} \cup c up _{n} D _{n,} which is a countable set (because it is a union of countably many countable sets) so that its subset D must also be countable (because every subset of a countable set is countable).

In particular, because every interval (including open intervals and half open/closed intervals) 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

f is an interval I that 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 = \cup ^{inf ty} _{n =1} I _{n.} If I = (a,b] with a \geq -inf ty then I _{1} = \left[\alpha _{1,} b\right], I _{2} = \left[\alpha _{2,} \alpha _{1\right],} \ldots, I _{n} = \left[\alpha _{n,} \alpha _{n-1} \right], \ldots where \left(\alpha _{n\right)} ^{inf ty} ^{n =1} is a strictly decreasing sequence such that \alpha _{n} → a. In a similar way if I = [a,b), with b \leq +inf ty or if I = (a,b) with -inf ty \leq a < b \leq inf ty. In any interval I _{n,} 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.\blacksquare

Jump functions

Examples. Let ₁ < ₂ < ₃ < ⋅⋅⋅ be a countable subset of the compact interval [{{mvar|''a''}},{{mvar|''b''}}] and let μ₁, μ₂, μ₃, ... be a positive sequence with finite sum. Set

f(x) = \sum ^{inf ty} _{n =1} \mu _{n} \chi _{[x _{n,b]}} (x)

where χ_A denotes the characteristic function of a compact interval . Then is a non-decreasing function on [{{mvar|''a''}},{{mvar|''b''}}], 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 [{{mvar|''a''}},{{mvar|''b''}}] 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 (≥ 1) lie in and take λ₁, λ₂, λ₃, ... and μ₁, μ₂, μ₃, ... non-negative with finite sum and with λ + μ > 0 for each . Define

f _{n(x)=0} for x < x _{n,} f _{n(x} _{n)} = λ _{n,} f _{n(x)} = λ _{n} + \mu _{n} for x > x _{n.}

Then the jump function, or saltus-function, defined by

f(x) = \sum ^{inf ty} _{n =1} f _{n(x)} = \sum _{x _{n\le} x} λ _{n} + \sum _{x _{n<x}} \mu _{n,}

is non-decreasing on [{{mvar|''a''}}, {{mvar|''b''}}] 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

f(x _{n)-f(x} _{n-0)=λ} _{n,} f(x _{n+0)-f(x} _{n)=\mu} _{n,} and f(x\pm 0 ) = f(x)

if is not one of the 's.

Conversely, by a differentiation theorem of Lebesgue, the jump function is uniquely determined by the properties:^[3] (1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity ; (3) satisfying the boundary condition = 0; and (4) having zero derivative almost everywhere.

Property (4) can be checked following, and . Without loss of generality, it can be assumed that is a non-negative jump function defined on the compact [{{mvar|''a''}},{{mvar|''b''}}], 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

{f(t) - f(s)\over t - s} > c

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.^[4] Hence

\ell(J _{k)} \le \sum _{m} (t _{k,m} - s _{k,m} ) \le \sum _{m} c ^{-1} (f(t _{k,m} ) - f(s _{k,m} )) \le 2 c ^{-1} (f(b _{k)-f(a} _{k)).} Finally sum both sides over .

Proposition 2. If is a jump function, then ' = 0 almost everywhere.

To prove this, define

Df(x) = \limsup _{s,t →} {f(t) - f(s)\over t - s},

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.

See also

• • • Monotone function

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Notes and References

1. Alexandre. Froda. Sur la distribution des propriétés de voisinage des functions de variables réelles. Hermann. Paris. 3 December 1929. 55.0742.02.
2. [Jean Gaston Darboux]
3. For more details, see
4. This is a simple example of how Lebesgue covering dimension applies in one real dimension; see for example .