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

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

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.

See also

• Continuous function
• Classification of discontinuities

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 Analysis (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 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).