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I am writing something about the following two theorems:

  1. Every function $f: \Bbb{R} \to \Bbb{R}$ can be written $f=f_1+f_2$ where $f_1,f_2:\Bbb{R} \to \Bbb{R}$ both have the Darboux property.

  2. Denote (C) the Cauchy functional equation: $$ f: \Bbb{R} \to \Bbb{R}, \ f(x+y)=f(x)+f(y)$$ Prove that every solution of (C) can be written $f=f_1+f_2$ where $f_1,f_2$ are (discontinuous) solutions of (C) which have the Darboux property.

How can I find the papers where the original proofs of these theorem first appeared. I tried googling, but Wikipedia doesn't work today. If the original articles aren't available, then maybe there are some books which contain the proof of the first theorem; those are good also. I only know a Romanian reference where the proof appears, but I would like to know a known English book which contains the proof. Thank you.

I found the second theorem in a problem book, and maybe it is more recent than the first one.

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Beni Bogosel
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2 Answers2

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I followed advice given by N.S. and looked what A. M. Bruckner says about the origin of the first theorem from your question. (I still don't have anything to say about the second one.) However, I've noticed that you've found a different paper by Sierpiński which you mention in your paper.

Andrew M. Bruckner. Differentiation of Real Functions (LNM659, Springer, 1978). BTW there is also a newer edition of this book, but both editions say the same thing regarding the above theorem, see here.

Theorem 4.1. Let $f$ be an arbitrary function on $\mathbb R$. There exist two Darboux functions $g$ and $h$ such that $f=g+h$.

Theorem 4.2. Let $f$ be an arbitrary function on $\mathbb R$. There exists a sequence $\{f_n\}$ of Darboux functions converging pointwise to $f$.

Theorems 4.1 and 4.2 were first announced by Lindenbaum [121]. A clever proof can be found in Fast [59].

A. M. Bruckner and J. Ceder: On the Sum of Darboux Functions; Proceedings of the American Mathematical Society , Vol. 51, No. 1 (Aug., 1975), pp. 97-102 jstor.

In recent years a number of articles have dealt with questions concerning the possible outcomes of adding two real functions with the Darboux property (i.e. the intermediate value property). For example, Sierpiński [8] (see also Fast [4]) showed that in the absence of further conditions on the functions, every function is the sum of two such (Darboux) functions.

The articles referenced in the above excerpts are:

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Martin Sleziak
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  • Thank you very much for your answer. – Beni Bogosel Apr 28 '12 at 09:17
  • @Beni I've seen the result that there are solutions of Cauchy equation such that $f[I]=\mathbb R$ for each interval, which is an interesting fact and the proof is nice. Didn't you consider posting this proof or link to your paper here or here. – Martin Sleziak May 09 '12 at 15:33
  • I thought my result was new, but I found some time ago essentially the same construction in a paper from the 60s of Bruckner. – Beni Bogosel May 09 '12 at 18:56
  • And also, I think that one of the answers on the MathOverflow presents the proof in my article. – Beni Bogosel May 09 '12 at 19:01
  • You mean Jim Belk's answer, right? His function is also additive and constant on every coset $x+\mathbb Q$. (I did not notice that these constructions are related until you pointed it out.) – Martin Sleziak May 10 '12 at 04:44
  • Yes, that is what I meant. Anyway, at the time I've written that article I was in my second year of college, about 3-4 years ago. Back then I thought I discoered something new, and the people I've asked about it told me also that the work I've done was new. I searched for references about the subject, but didn't managed to find some similar articles until this year when I read Bruckner's article and the book. – Beni Bogosel May 10 '12 at 09:50
  • Ok, I made a mistake above, so here's a correction: Function obtained from Jim Belk's solution are not necessarily additive. They can be made additive if we choose a suitable bijection between $\mathbb R/\mathbb Q$ and $\mathbb R$. However, to find such bijection we would probaly use Hamel basis, so this would be basically a reformulation of your approach. – Martin Sleziak May 22 '12 at 05:47
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Wikipedia provides a reference for the first theorem:

Bruckner, Andrew M: Differentiation of real functions, 2 ed, page 6, American Mathematical Society, 1994

Probably not the best approach, but you could check that book and try to backtrack the Theorem.

Martin Sleziak
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N. S.
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