Froda's theorem

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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 A. Froda' thesis in 1929 .[1][2][dubious ]. As it is acknowledged in the thesis, it is in fact due Jean Gaston Darboux [3]


Definitions[edit]

  1. Consider a function f of real variable x with real values defined in a neighborhood of a point x_0 and the function f is discontinuous at the point on the real axis x = x_0. We will call a removable discontinuity or a jump discontinuity a discontinuity of the first kind.[4]
  2. Denote f(x+0):=\lim_{h\searrow0}f(x+h) and f(x-0):=\lim_{h\searrow0}f(x-h). Then if f(x_0+0) and f(x_0-0) are finite we will call the difference f(x_0+0)-f(x_0-0) the jump[5] of f at x_0.

If the function is continuous at x_0 then the jump at x_0 is zero. Moreover, if f is not continuous at x_0, the jump can be zero at x_0 if f(x_0+0)=f(x_0-0)\neq f(x_0).

Precise statement[edit]

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.

Proof[edit]

Let I:=[a,b] be an interval and f defined on I an increasing function. We have

f(a)\leq f(a+0)\leq f(x-0)\leq f(x+0)\leq f(b-0)\leq f(b)

for any a<x<b. Let \alpha >0 and let x_1<x_2<\cdots<x_n be n points inside I at which the jump of f is greater or equal to \alpha:

f(x_i+0)-f(x_i-0)\geq \alpha,\ i=1,2,\ldots,n

We have f(x_i+0)\leq f(x_{i+1}-0) or f(x_{i+1}-0)-f(x_i+0)\geq 0,\ i=1,2,\ldots,n. Then

f(b)-f(a)\geq f(x_n+0)-f(x_1-0)=\sum_{i=1}^n [f(x_i+0)-f(x_i-0)]+
+\sum_{i=1}^{n-1}[f(x_{i+1}-0)-f(x_i+0)]\geq \sum_{i=1}^n[f(x_i+0)-f(x_i-0)]\geq n\alpha

and hence: n\leq \frac{f(b)-f(a)}{\alpha}\ .

Since f(b)-f(a) <\infty we have that the number of points at which the jump is greater than \alpha is finite or zero.

We define the following sets:

S_1:=\{x:x\in I, f(x+0)-f(x-0)\geq 1\},
S_n:=\{x:x\in I, \frac{1}{n}\leq f(x+0)-f(x-0)<\frac{1}{n-1}\},\ n\geq 2.

We have that each set S_n is finite or the empty set. The union S=\cup_{n=1}^\infty 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, we have that S is at most countable.

If f is decreasing the proof is similar.

If the interval I 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_{n=1}^\infty I_n.

If I=(a,b],\ a\geq -\infty \ then I_1=[\alpha_1,b],\ I_2=[\alpha_2,\alpha_1],\ldots,\ I_n=[\alpha_n,\alpha_{n-1}],\ldots where \{\alpha_n\}_n is a strictly decreasing sequence such that \alpha_n\rightarrow a.\ In a similar way if I=[a,b),\ b\leq+\infty\ or if I=(a,b)\ -\infty\leq a<b\leq \infty.

In any interval I_n 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.

Remark[edit]

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 I. Then the set of discontinuities is at most countable.

See also[edit]

Notes[edit]

  1. ^ Alexandru Froda, Sur la Distribution des Propriétés de Voisinage des Fonctions de Variables Réelles, These, Harmann, Paris, 3 December 1929
  2. ^ Alexandru Froda – Collected Papers (Opera Matematica), Vol.1, Ed. Academ. Romane, 2000
  3. ^ Jean Gaston Darboux Mémoire sur les fonctions discontinues, Annales 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. ^ M. Nicolescu, N. Dinculeanu, S. Marcus, Mathematical Anlaysis (Bucharest 1971), Vol.1, Pg.213, [in Romanian]
  6. ^ W. Rudin, Principles of Mathematical Analysis, McGraw–Hill 1964 (Corollary, p.83)
  7. ^ M. Nicolescu, N. Dinculeanu, S. Marcus, Mathematical Anlaysis (Bucharest 1971), Vol.1, Pg.213, [in Romanian]

References[edit]

  • 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).