Froda's theorem

In mathematics, Darboux–Froda's theorem, named after Alexandru Froda, a Romanian mathematician, describes the set of discontinuities of a monotone real-valued function of a real variable. Usually, this theorem appears in literature without a name. It was written in Froda' thesis in 1929.[1][2][dubious ]. As it is acknowledged in the thesis, the theorem is in fact due to Jean Gaston Darboux.[3]

DefinitionsEdit

  1. Consider a function f of real variable x with real values defined in a neighborhood of a point   and the function f 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.[4]
  2. Denote   and  . Then if   and   are finite we will call the difference   the jump[5] of f at  .

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 statementEdit

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

ProofEdit

Let   be an interval and  , defined on  , an increasing 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. 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 the proof is similar.

If the interval   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   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.

See alsoEdit

NotesEdit

  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. ^ Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill 1964, (Def. 4.26, pp. 81–82)
  5. ^ Miron Nicolescu, Nicolae Dinculeanu, Solomon Marcus, Mathematical Analysis (Bucharest 1971), Vol. 1, p. 213, [in Romanian]
  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]

ReferencesEdit

  • 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).