Froda's theorem

Summary

In mathematics, Darboux–Froda's 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 was written in the thesis of Romanian mathematician Alexandru Froda in 1929.[1][2][dubious ]. As it is acknowledged in the thesis, the theorem is in fact due to French mathematician Jean Gaston Darboux.[3]

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[4] 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).[5] 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[6][7] 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 Froda's 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 (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

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

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[note 1]). 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.

See also

Notes

  1. ^ Alexandre Froda, Sur la Distribution des Propriétés de Voisinage des Fonctions de Variables Réelles, Thèse, Éditions Hermann, Paris, 3 December 1929
  2. ^ Alexandru Froda – Collected Papers (Opera Matematica), Vol.1, Editor Academiei Române, 2000
  3. ^ Jean Gaston Darboux, Mémoire sur les fonctions discontinues, Annales Scientifiques de l'École Normale Supérieure, 2-ème série, t. IV, 1875, Chap VI.
  4. ^ Miron Nicolescu, Nicolae Dinculeanu, Solomon Marcus, Mathematical Analysis (Bucharest 1971), Vol. 1, p. 213, [in Romanian]
  5. ^ Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill 1964, (Def. 4.26, pp. 81–82)
  6. ^ Walter Rudin, Principles of Mathematical Analysis, McGraw–Hill 1964 (Corollary, p. 83)
  7. ^ Miron Nicolescu, Nicolae Dinculeanu, Solomon Marcus, Mathematical Analysis (Bucharest 1971), Vol.1, p. 213, [in Romanian]
  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

  • Bernard R. Gelbaum, John M. H. Olmsted, Counterexamples in Analysis, Holden–Day, Inc., 1964. (18. Page 28)
  • John M. H. Olmsted, Real Variables, Appleton–Century–Crofts, Inc., New York (1956), (Page 59, Ex. 29).