I am reading on Froda's Theorem and I am trying to understand the proof given on this page. I follow up until the sets are defined:
$S_1:= \{x: x\in I, f(x+0)-f(x-0)\ge 1 \}$
$S_n:= \{x: x\in I, \frac{1}{n}\le f(x+0)-f(x-0)< \frac{1}{n-1} \},n\ge 2$
I fail to see why $f(x+0)-f(x-0)\ge 1$ and where does the $\frac{1}{n}$ and $\frac{1}{n-1}$ come from?
Here is a minimal answer along with a clarification in case someone else happens to find the mysterious Froda here.
The theorem you are trying to prove is that the set of jump discontinuities of a monotone nondecreasing function $f:[a,b]\to R$ is countable. At each point $x$ you measure the jump size as $f(x+0)-f(x-0)$. Collect all the jumps of size greater than $1/n$. Since the function grows only to the value $f(b)$ there cannot be infinitely many such jumps. If $J_n$ is the set of all points $x$ in the interval with a jump greater than $1/n$ then the set $$J = \bigcup_{n=1}^\infty J_n$$ contains all points at which there is a jump.
But this theorem was known long before Froda and it was not the subject of his 1929 thesis anyway. He proved that for a completely arbitrary function [not necessarily monotone] the set of points at which there is a removable or a jump discontinuity is countable. And not by this method.
And ... he has no priority on this either. He first reproduces a proof of a theorem of Darboux. That proves that if you assume a function $f$ has no essential discontinuities (i.e., has only removable or jump discontinuities) then the set of discontinuities is countable. Froda's thesis then goes on to show this for a general function: even if you allow essential discontinuities the collection of jump/removable discontinuities is countable.
But if we go back much earlier to 1908 we find W. H. Young has proved what he called his "Rome theorem" (because he went to the international meeting and presented it there). That theorem may be one of the first theorems about completely arbitrary functions. (Before then a typical theorem would assume something about a function and then prove something else.) His theorem includes all of this and much more. I won't intrude with the details but roughly speaking he says that for any function there are only countably many points where there is this kind of asymmetry between the left and the right.
I haven't read all of Froda's thesis (it's in French which causes me even more pain now than it did years ago) but it is a legitimate contribution. On this topic, however, he has no priority and it was a shock to see his name associated with that elementary theorem.
[Postscript: There is a bit of an Alexandru Froda (1894–1973) mystery here. The Wiki entry on him seems to be quite confused as to what he did and whether he has priority on it. I haven't the patience to read any more of his 111 page thesis and I don't want to be a Wiki editor. The original poster, I gather, was using Wiki and trying to learn from that. The mystery deepens since the Math Geneology entry says that his supervisor is unknown but he was awarded a Ph.D. at Université Paris IV-Sorbonne in 1929. No known students. The examining committe was apparently Borel, Montel and Denjoy. Pompeiu (who was a Romanian like Froda) was perhaps the supervisor. If anyone can add something, please do.]
