# 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 .{{ 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 

## 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

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

One can prove 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.