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I heard this problem and I am a bit stuck.

Given a function $f : I \rightarrow \mathbb{R}$ where $I \subset \mathbb{R}$ is an open interval. Then $f$ can be writen $f=g+h$ where $g,h$ are defined in the same interval and have the Intermediate Value Property. I tried to construct firstly the one function arbitarily at two points and then tried to define it in a way to have the IVP but I cannot manage to control the other function, as I try to fix the one I destroy the other and I cannot seem to know how to be certain I have enough point to define both in a way they have the IVP.

Any help appreciated! Thank you.

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Theorem 9.5, p.57 in van Rooij, Schikhof: A Second Course on Real Functions. –  Martin Sleziak Aug 6 '12 at 4:41
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up vote 4 down vote accepted

Edit: In fact, all the information I give below (and more) is provided in another question in a much more organized way. I just found it.

My original post: The intermediate Value property is also called the Darboux property. Sierpinski first proved this theorem.The problem is treated in a blog of Beni Bogosel, a member of our own community and in much more generality too.

http://mathproblems123.files.wordpress.com/2010/07/strange-functions.pdf

It is also proved in( As I found from Wikipedia)

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

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This is funny, because you linked to a blog of a member of MSE. –  mixedmath Aug 6 '12 at 2:01
    
@mixedmath Thank you! Took me a moment to realize. :) –  Ravi Donepudi Aug 6 '12 at 2:07
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Thanks very much for the interesting link, and very elementary! Actually I knew about mr. Bogosel's blog he is doing very good work there. –  clark Aug 6 '12 at 2:17
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