# Froda's theorem

{{ safesubst:#invoke:Unsubst||$N=Use dmy dates |date=__DATE__ |$B= }} 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]{{ safesubst:#invoke:Unsubst||$N=Dubious |date=__DATE__ |$B= {{#invoke:Category handler|main}}[dubious ] }}. As it is acknowledged in the thesis, it is in fact due Jean Gaston Darboux [3]

## Definitions

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

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

## 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.

## Proof

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

${\displaystyle f(a)\leq f(a+0)\leq f(x-0)\leq f(x+0)\leq f(b-0)\leq f(b)}$
${\displaystyle f(x_{i}+0)-f(x_{i}-0)\geq \alpha ,\ i=1,2,\ldots ,n}$
${\displaystyle f(b)-f(a)\geq f(x_{n}+0)-f(x_{1}-0)=\sum _{i=1}^{n}[f(x_{i}+0)-f(x_{i}-0)]+}$
${\displaystyle +\sum _{i=1}^{n-1}[f(x_{i+1}-0)-f(x_{i}+0)]\geq \sum _{i=1}^{n}[f(x_{i}+0)-f(x_{i}-0)]\geq n\alpha }$

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

We define the following sets:

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

We have that each set ${\displaystyle S_{n}}$ is finite or the empty set. The union ${\displaystyle S=\cup _{n=1}^{\infty }S_{n}}$ contains all points at which the jump is positive and hence contains all points of discontinuity. Since every ${\displaystyle S_{i},\ i=1,2,\ldots \ }$ is at most countable, we have that ${\displaystyle S}$ is at most countable.

If ${\displaystyle f}$ is decreasing the proof is similar.

If the interval ${\displaystyle I}$ 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 ${\displaystyle I_{n}}$ with the property that any two consecutive intervals have an endpoint in common: ${\displaystyle I=\cup _{n=1}^{\infty }I_{n}.}$

In any interval ${\displaystyle I_{n}}$ 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

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 ${\displaystyle I}$. Then the set of discontinuities is at most countable.

## Notes

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

• 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).