Function writen as two functions having IVP 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.

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
