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.. As it is acknowledged in the thesis, the theorem is in fact due to Jean Gaston Darboux.

Definitions

1. Consider a function f of real variable x with real values defined in a neighborhood of a point [math]\displaystyle{ x_0 }[/math] and the function f is discontinuous at the point on the real axis [math]\displaystyle{ x = x_0 }[/math]. We will call a removable discontinuity or a jump discontinuity a discontinuity of the first kind.
2. Denote [math]\displaystyle{ f(x+0):=\lim_{h\searrow0}f(x+h) }[/math] and [math]\displaystyle{ f(x-0):=\lim_{h\searrow0}f(x-h) }[/math]. Then if [math]\displaystyle{ f(x_0+0) }[/math] and [math]\displaystyle{ f(x_0-0) }[/math] are finite we will call the difference [math]\displaystyle{ f(x_0+0)-f(x_0-0) }[/math] the jump of f at [math]\displaystyle{ x_0 }[/math].

If the function is continuous at [math]\displaystyle{ x_0 }[/math] then the jump at [math]\displaystyle{ x_0 }[/math] is zero. Moreover, if [math]\displaystyle{ f }[/math] is not continuous at [math]\displaystyle{ x_0 }[/math], the jump can be zero at [math]\displaystyle{ x_0 }[/math] if [math]\displaystyle{ f(x_0+0)=f(x_0-0)\neq f(x_0) }[/math].

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 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 [math]\displaystyle{ I }[/math]. Then the set of discontinuities is at most countable.

Proof

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

[math]\displaystyle{ f(a)\leq f(a+0)\leq f(x-0)\leq f(x+0)\leq f(b-0)\leq f(b) }[/math]

for any [math]\displaystyle{ a\lt x\lt b }[/math]. Let [math]\displaystyle{ \alpha \gt 0 }[/math] and let [math]\displaystyle{ x_1\lt x_2\lt \cdots\lt x_n }[/math] be [math]\displaystyle{ n }[/math] points inside [math]\displaystyle{ I }[/math] at which the jump of [math]\displaystyle{ f }[/math] is greater or equal to [math]\displaystyle{ \alpha }[/math]:

[math]\displaystyle{ f(x_i+0)-f(x_i-0)\geq \alpha,\ i=1,2,\ldots,n }[/math]

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

[math]\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)]+ }[/math]

[math]\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 }[/math]

and hence: [math]\displaystyle{ n\leq \frac{f(b)-f(a)}{\alpha} }[/math].

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

We define the following sets:

[math]\displaystyle{ S_1:=\{x:x\in I, f(x+0)-f(x-0)\geq 1\} }[/math],

[math]\displaystyle{ S_n:=\{x:x\in I, \frac{1}{n}\leq f(x+0)-f(x-0)\lt \frac{1}{n-1}\},\ n\geq 2. }[/math]

We have that each set [math]\displaystyle{ S_n }[/math] is finite or the empty set. The union [math]\displaystyle{ S=\bigcup_{n=1}^\infty S_n }[/math] contains all points at which the jump is positive and hence contains all points of discontinuity. Since every [math]\displaystyle{ S_i,\ i=1,2,\ldots }[/math] is at most countable, we have that [math]\displaystyle{ S }[/math] is at most countable.

If [math]\displaystyle{ f }[/math] is decreasing the proof is similar.

If the interval [math]\displaystyle{ I }[/math] 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 [math]\displaystyle{ I_n }[/math] with the property that any two consecutive intervals have an endpoint in common: [math]\displaystyle{ I=\cup_{n=1}^\infty I_n. }[/math]

If [math]\displaystyle{ I=(a,b],\ a\geq -\infty }[/math] then [math]\displaystyle{ I_1=[\alpha_1,b],\ I_2=[\alpha_2,\alpha_1],\ldots,\ I_n=[\alpha_n,\alpha_{n-1}],\ldots }[/math] where [math]\displaystyle{ \{\alpha_n\}_n }[/math] is a strictly decreasing sequence such that [math]\displaystyle{ \alpha_n\rightarrow a. }[/math] In a similar way if [math]\displaystyle{ I=[a,b),\ b\leq+\infty }[/math] or if [math]\displaystyle{ I=(a,b)\ -\infty\leq a\lt b\leq \infty }[/math].

In any interval [math]\displaystyle{ I_n }[/math] 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

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).