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]. As it is acknowledged in the thesis, the theorem is in fact due to 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.

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.

Proof

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

${\displaystyle f(a)\leq f(a+0)\leq f(x-0)\leq f(x+0)\leq f(b-0)\leq f(b)}$

for any ${\displaystyle a<x<b}$. Let ${\displaystyle \alpha >0}$ and let ${\displaystyle x_{1}<x_{2}<\cdots <x_{n}}$ be ${\displaystyle n}$ points inside ${\displaystyle I}$ at which the jump of ${\displaystyle f}$ is greater or equal to ${\displaystyle \alpha }$:

${\displaystyle f(x_{i}+0)-f(x_{i}-0)\geq \alpha ,\ i=1,2,\ldots ,n}$

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

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

${\displaystyle +\sum _{i=1}^{n-1}[f(x_{i+1}-0)-f(x_{i}+0)]\geq \sum _{i=1}^{n}[f(x_{i}+0)-f(x_{i}-0)]\geq n\alpha }$

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

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 ${\displaystyle S=\bigcup _{n=1}^{\infty }S_{n}}$ 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: ${\displaystyle I=\cup _{n=1}^{\infty }I_{n}.}$

If ${\displaystyle I=(a,b],\ a\geq -\infty }$ then ${\displaystyle I_{1}=[\alpha _{1},b],\ I_{2}=[\alpha _{2},\alpha _{1}],\ldots ,\ I_{n}=[\alpha _{n},\alpha _{n-1}],\ldots }$ where ${\displaystyle \{\alpha _{n}\}_{n}}$ is a strictly decreasing sequence such that ${\displaystyle \alpha _{n}\rightarrow a.}$ In a similar way if ${\displaystyle I=[a,b),\ b\leq +\infty }$ or if ${\displaystyle I=(a,b)\ -\infty \leq a<b\leq \infty }$.

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.

See also

• Continuous function
• Classification of discontinuities

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. 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]

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).