Discontinuities of monotone functions

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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

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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[3] 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 (also called a discontinuity of the first kind).[4] 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

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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[5][3] 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

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This proof starts by proving the special case where the function's domain is a closed and bounded interval  [6][7] The proof of the general case follows from this special case.

Proof when the domain is closed and bounded

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Two proofs of this special case are given.

Proof 1

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Let   be an interval and let   be a non-decreasing function (such as an increasing 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 (possibly even zero).

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 (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.  

Proof 2

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So let   is a monotone function and let   denote the set of all points   in the domain of   at which   is discontinuous (which is necessarily a jump discontinuity).

Because   has a jump discontinuity at     so there exists some rational number   that lies strictly in between   (specifically, if   then pick   so that   while if   then pick   so that   holds).

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 (said differently, the map   defined by   is injective). Since   is countable, the same must be true of    

Proof of general case

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Suppose that the domain of   (a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is   (no requirements are placed on these closed and bounded intervals[a]). 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 (countable) 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 (that is,  ) or else there exists some index   such that   in which case   must be a point of discontinuity for   (that is,  ). Thus the set   of all points of at which   is discontinuous is a subset of   which is a countable set (because it is a union of countably many countable sets) so that its subset   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   is an interval   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   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

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Examples. Let x1 < x2 < x3 < ⋅⋅⋅ be a countable subset of the compact interval [a,b] and let μ1, μ2, μ3, ... be a positive sequence with finite sum. Set

 

where χA denotes the characteristic function of a compact interval A. Then f is a non-decreasing function on [a,b], which is continuous except for jump discontinuities at xn for n ≥ 1. In the case of finitely many jump discontinuities, f 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.[8][9]

More generally, the analysis of monotone functions has been studied by many mathematicians, starting from Abel, Jordan and Darboux. Following Riesz & Sz.-Nagy (1990), replacing a function by its negative if necessary, only the case of non-negative non-decreasing functions has to be considered. The domain [a,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 xn (n ≥ 1) lie in (a, b) and take λ1, λ2, λ3, ... and μ1, μ2, μ3, ... non-negative with finite sum and with λn + μn > 0 for each n. Define

  for   for  

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

 

is non-decreasing on [a, b] and is continuous except for jump discontinuities at xn for n ≥ 1.[10][11][12][13]

To prove this, note that sup |fn| = λn + μn, so that Σ fn converges uniformly to f. Passing to the limit, it follows that

  and  

if x is not one of the xn's.[10]

Conversely, by a differentiation theorem of Lebesgue, the jump function f is uniquely determined by the properties:[14] (1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity xn; (3) satisfying the boundary condition f(a) = 0; and (4) having zero derivative almost everywhere.

Proof that a jump function has zero derivative almost everywhere.

Property (4) can be checked following Riesz & Sz.-Nagy (1990), Rubel (1963) and Komornik (2016). Without loss of generality, it can be assumed that f is a non-negative jump function defined on the compact [a,b], with discontinuities only in (a,b).

Note that an open set U of (a,b) is canonically the disjoint union of at most countably many open intervals Im; that allows the total length to be computed ℓ(U)= Σ ℓ(Im). Recall that a null set A is a subset such that, for any arbitrarily small ε' > 0, there is an open U containing A with ℓ(U) < ε'. A crucial property of length is that, if U and V are open in (a,b), then ℓ(U) + ℓ(V) = ℓ(UV) + ℓ(UV).[15] It implies immediately that the union of two null sets is null; and that a finite or countable set is null.[16][17]

Proposition 1. For c > 0 and a normalised non-negative jump function f, let Uc(f) be the set of points x such that

 

for some s, t with s < x < t. Then Uc(f) is open and has total length ℓ(Uc(f)) ≤ 4 c−1 (f(b) – f(a)).

Note that Uc(f) consists the points x where the slope of h is greater that c near x. By definition Uc(f) is an open subset of (a, b), so can be written as a disjoint union of at most countably many open intervals Ik = (ak, bk). Let Jk be an interval with closure in Ik and ℓ(Jk) = ℓ(Ik)/2. By compactness, there are finitely many open intervals of the form (s,t) covering the closure of Jk. 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 (sk,1,tk,1), (sk,2,tk,2), ... only intersecting at consecutive intervals.[18] Hence

 

Finally sum both sides over k.[16][17]

Proposition 2. If f is a jump function, then f '(x) = 0 almost everywhere.

To prove this, define

 

a variant of the Dini derivative of f. It will suffice to prove that for any fixed c > 0, the Dini derivative satisfies Df(x) ≤ c almost everywhere, i.e. on a null set.

Choose ε > 0, arbitrarily small. Starting from the definition of the jump function f = Σ fn, write f = g + h with g = ΣnN fn and h = Σn>N fn where N ≥ 1. Thus g is a step function having only finitely many discontinuities at xn for nN and h is a non-negative jump function. It follows that Df = g' +Dh = Dh except at the N points of discontinuity of g. Choosing N sufficiently large so that Σn>N λn + μn < ε, it follows that h is a jump function such that h(b) − h(a) < ε and Dhc off an open set with length less than 4ε/c.

By construction Dfc off an open set with length less than 4ε/c. Now set ε' = 4ε/c — then ε' and c are arbitrarily small and Dfc off an open set of length less than ε'. Thus Dfc almost everywhere. Since c could be taken arbitrarily small, Df and hence also f ' must vanish almost everywhere.[16][17]

As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function F can be decomposed uniquely as a sum of a jump function f and a continuous monotone function g: the jump function f is constructed by using the jump data of the original monotone function F and it is easy to check that g = Ff is continuous and monotone.[10]

See also

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Notes

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  1. ^ So for instance, these intervals need not be pairwise disjoint nor is it required that they intersect only at endpoints. It is even possible that   for all  

References

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  1. ^ Froda, Alexandre (3 December 1929). Sur la distribution des propriétés de voisinage des functions de variables réelles (PDF) (Thesis). Paris: Hermann. JFM 55.0742.02.
  2. ^ Jean Gaston Darboux, Mémoire sur les fonctions discontinues, Annales Scientifiques de l'École Normale Supérieure, 2-ème série, t. IV, 1875, Chap VI.
  3. ^ a b Nicolescu, Dinculeanu & Marcus 1971, p. 213.
  4. ^ Rudin 1964, Def. 4.26, pp. 81–82.
  5. ^ Rudin 1964, Corollary, p. 83.
  6. ^ Apostol 1957, pp. 162–3.
  7. ^ Hobson 1907, p. 245.
  8. ^ Apostol 1957.
  9. ^ Riesz & Sz.-Nagy 1990.
  10. ^ a b c Riesz & Sz.-Nagy 1990, pp. 13–15
  11. ^ Saks 1937.
  12. ^ Natanson 1955.
  13. ^ Łojasiewicz 1988.
  14. ^ For more details, see
  15. ^ Burkill 1951, pp. 10−11.
  16. ^ a b c Rubel 1963
  17. ^ a b c Komornik 2016
  18. ^ This is a simple example of how Lebesgue covering dimension applies in one real dimension; see for example Edgar (2008).

Bibliography

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