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

Definitions

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]
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

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

Proof

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.

Notes

↑ Alexandru Froda, Sur la Distribution des Propriétés de Voisinage des Fonctions de Variables Réelles, These, Harmann, Paris, 3 December 1929
↑ Alexandru Froda – Collected Papers (Opera Matematica), Vol.1, Ed. Academ. Romane, 2000
↑ Jean Gaston Darboux Mémoire sur les fonctions discontinues, Annales de l'École Normale supérieure, 2-ème série, t. IV, 1875, Chap VI.
↑ Walter Rudin, Principles of Mathematical Analysis, McGraw-Hill 1964, (Def. 4.26, pp. 81–82)
↑ M. Nicolescu, N. Dinculeanu, S. Marcus, Mathematical Analysis (Bucharest 1971), Vol.1, Pg.213, [in Romanian]
↑ W. Rudin, Principles of Mathematical Analysis, McGraw–Hill 1964 (Corollary, p.83)
↑ M. Nicolescu, N. Dinculeanu, S. Marcus, Mathematical Analysis (Bucharest 1971), Vol.1, Pg.213, [in Romanian]

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).

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