My Math Forum integrability

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 January 20th, 2011, 09:13 AM #1 Member   Joined: Oct 2010 Posts: 37 Thanks: 0 integrability Show that there does not exist any function $\phi$ such that $\phi'(x)=f(x)$ ,where (i). $f(x)=[x], 0\leq x\leq 2$ (ii).$f(x)=x-[x], 0\leq x\leq 2$ We know that box-functions are well-integrable. How can the above hold then?
 January 20th, 2011, 03:41 PM #2 Global Moderator   Joined: May 2007 Posts: 6,510 Thanks: 584 Re: integrability If I understand the problem correctly, the requirement is equality at every point. At the discontinuity points of f(x), ?'(x) will not be defined.
January 21st, 2011, 04:29 AM   #3
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Re: integrability

Quote:
 Originally Posted by mathman requirement is equality at every point
what is that?

 January 21st, 2011, 05:11 AM #4 Senior Member   Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3 Re: integrability $\phi ' (1) = \lim_{h\to 0} \frac{\phi(1+h)-\phi(1)}{h}$ but does the limit exist?
 January 22nd, 2011, 06:08 AM #5 Member   Joined: Oct 2010 Posts: 37 Thanks: 0 Re: integrability No. In this way the problem is solved though. But it does not deal with integrability whatsoever. Whatever. Thanks.
January 22nd, 2011, 03:09 PM   #6
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Re: integrability

Quote:
Originally Posted by Sambit
Quote:
 Originally Posted by mathman requirement is equality at every point
what is that?
?'(x)=f(x) for all x, 0?x?2.

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