Froda's theorem

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.
2. Denote f ( x + 0 ) := lim h ↘ 0 f ( x + h ) and f ( x − 0 ) := lim h ↘ 0 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 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 ) ≠ 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 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

for any a < x < b . Let α > 0 and let x 1 < x 2 < ⋯ < x n be n points inside I at which the jump of f is greater or equal to α :

We have f ( x i + 0 ) ≤ f ( x i + 1 − 0 ) or f ( x i + 1 − 0 ) − f ( x i + 0 ) ≥ 0 ,   i = 1 , 2 , … , n . Then

and hence: n ≤ f ( b ) − f ( a ) α   .

Since f ( b ) − f ( a ) < ∞ we have that the number of points at which the jump is greater than α is finite or zero.

We define the following sets:

We have that each set S n is finite or the empty set. The union S = ∪ n = 1 ∞ 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 , …   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 = ∪ n = 1 ∞ I n .

If I = ( a , b ] ,   a ≥ − ∞   then I 1 = [ α 1 , b ] ,   I 2 = [ α 2 , α 1 ] , … ,   I n = [ α n , α n − 1 ] , … where { α n } n is a strictly decreasing sequence such that α n → a .   In a similar way if I = [ a , b ) ,   b ≤ + ∞   or if I = ( a , b )   − ∞ ≤ a < b ≤ ∞ .

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.