From Wikipedia: Darboux functions are a quite general class of functions. It turns out that any real-valued function f on the real line can be written as the sum of two Darboux functions. This implies in particular that the class of Darboux functions is not closed under addition.

(Darboux functions are simply those that satisfy the intermediate value property).

Proof? I'm looking for this out of interest, and couldn't find one - a hint, nudge, reference or link are also sufficient. If you're feeling brave: can we extend this, say to $f:\mathbb{C} \to \mathbb{R}$? Proof or counter-example, of course.

EDIT: Further question: If a function $f:[a,b] \to \mathbb{R}$ is differentiable on $[a,b]$ then its derivative $f'$ is Darboux on $[a,b]$. Can any real function be written as $f'+g'$ for some $f,g$? (no)

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    $\begingroup$ See this. For the edit, the answer is no. Otherwise, every function would be a derivative, which isn't true. $\endgroup$ – David Mitra Sep 27 '14 at 9:11
  • $\begingroup$ Not to be an arse (well, ok, I guess precisely to be an arse), but let me google that for you. The very first PDF hit contains a reference to the paper by Sierpinski where the result is proven. $\endgroup$ – Jonas Dahlbæk Sep 27 '14 at 9:11
  • $\begingroup$ @user161825 I said a link/reference was fine and clearly I'd looked, just simply not well enough. No need to be an arse... $\endgroup$ – Shakespeare Sep 27 '14 at 9:13
  • $\begingroup$ Well, since I supplied you with what you wanted, I would only classify my comment as that of a half-arse. $\endgroup$ – Jonas Dahlbæk Sep 27 '14 at 9:16
  • $\begingroup$ @user161825: Have you read that paper?I came to this on the very first page: "Definition 2. If $f : I \to \mathbb R$ is a function, and $I \subset \mathbb R$ is an interval, $f$ has the Intermediate Value property if $f(I)$ is an interval." That looked like nonsense to me, so I gave up. $\endgroup$ – TonyK Sep 27 '14 at 9:17

Further question: $f' + g' = (f+g)'$ is a derivative and is therefore a Darboux function. So no, not every real function can be written as $f'+g'$.

  • $\begingroup$ I was just about to edit this into my question. Thanks though :) $\endgroup$ – Shakespeare Sep 27 '14 at 9:11

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