Function writen as two functions having IVP

Question

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.

Theorem 9.5, p.57 in van Rooij, Schikhof: A Second Course on Real Functions. — Martin Sleziak, Aug 06 '12 at 04:41
See also: http://math.stackexchange.com/questions/885580/is-every-real-valued-function-on-an-interval-a-sum-of-two-functions-with-interme — Martin Sleziak, Aug 06 '14 at 07:43

score 4 · Accepted Answer · edited Apr 13 '17 at 12:20

4

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

edited Apr 13 '17 at 12:20

Community

1

answered Aug 06 '12 at 01:59

Arkady

9,315

This is funny, because you linked to a blog of a member of MSE. – davidlowryduda Aug 06 '12 at 02:01
@mixedmath Thank you! Took me a moment to realize. :) – Arkady Aug 06 '12 at 02:07
1

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 06 '12 at 02:17

Function writen as two functions having IVP

1 Answers1

Linked