# Integrable functions.

• Feb 26th 2011, 08:02 AM
Boaz
Integrable functions.
Hi,
I need to find a function f, which is not integrable in [0,1], but the function |f| is integrable in that interval [0,1].

• Feb 26th 2011, 08:09 AM
Plato
Quote:

Originally Posted by Boaz
Hi,
I need to find a function f, which is not integrable in [0,1], but the function |f| is integrable in that interval [0,1].

$\displaystyle f(x) = \left\{ {\begin{array}{rl} {1,} & {x \in \left[ {0,1} \right] \cap \mathbb{Q}} \\ { - 1,} & {x \in \left[ {0,1} \right]\backslash \mathbb{Q}} \\ \end{array} } \right.$

What can you do with that and why?
• Feb 26th 2011, 08:15 AM
Boaz
Quote:

Originally Posted by Plato
$\displaystyle f(x) = \left\{ {\begin{array}{rl} {1,} & {x \in \left[ {0,1} \right] \cap \mathbb{Q}} \\ { - 1,} & {x \in \left[ {0,1} \right]\backslash \mathbb{Q}} \\ \end{array} } \right.$

What can you do with that and why?

Im not sure i understand why f isnt integrable..is it because the upper and the lower sums are very "far" from each other in every interval of the partition?
• Feb 26th 2011, 08:25 AM
Plato
Quote:

Originally Posted by Boaz
Im not sure i understand why f isnt integrable..is it because the upper and the lower sums are very "far" from each other in every interval of the partition?

Well, yes. That function is nowhere continuous.
• Feb 26th 2011, 08:31 AM
Boaz
Quote:

Originally Posted by Plato
Well, yes. That function is nowhere continuous.

oh right, I forgot about that!
Thank you very much!