# Example of function not integrable and the square is

• Jul 5th 2011, 01:13 AM
CountingPenguins
Example of function not integrable and the square is
I'm looking for the example of a function f:[0,1]->R that is not integrable on [0,1] but the square of the function is.

I've tried 1/x and lnx and e^x and a bunch of others, I'm thinking it could have something to do with a transformation of (-x^3) then when it's squared the function is x^6, but I'm not sure if this is the right track. Any ideas?
• Jul 5th 2011, 01:31 AM
CountingPenguins
Re: Example of function not integrable and the square is
I figured it out. The sin(1/x) integral does not exist and the integral of the square does?
• Jul 5th 2011, 01:32 AM
girdav
Re: Example of function not integrable and the square is
What does Cauchy-Schwarz inequality give?
edit : (do you mean Riemann or Lebesgue integrable?)
• Jul 5th 2011, 02:55 AM
Plato
Re: Example of function not integrable and the square is
Quote:

Originally Posted by CountingPenguins
I'm looking for the example of a function f:[0,1]->R that is not integrable on [0,1] but the square of the function is.

What about $\displaystyle f(x) = \left\{ {\begin{array}{rl} {1,} & {x \in \mathbb{Q}} \\ { - 1,} & {x \notin \mathbb{Q}} \\ \end{array} } \right.$
• Jul 5th 2011, 06:09 AM
Also sprach Zarathustra
Re: Example of function not integrable and the square is
Quote:

Originally Posted by Plato
What about $\displaystyle f(x) = \left\{ {\begin{array}{rl} {1,} & {x \in \mathbb{Q}} \\ { - 1,} & {x \notin \mathbb{Q}} \\ \end{array} } \right.$

Proof that $\displaystyle f(x)$ is not not integrable on $\displaystyle [0,1]$ (or on any closed interval $\displaystyle [a,b]$).

For every possible partition $\displaystyle P=\{\Delta {x_1},\Delta {x_2},...,\Delta {x_n}\}$ if we choose rational points $\displaystyle t_i\in\Delta {x_i}$, we'll get:

Riemann's sum: $\displaystyle \sum_{i=1}^{n}f(t_i)\Delta {x_i}=\sum_{i=1}^{n}1\Delta {x_i}=1-0=1$

Now, do the same for $\displaystyle k_i$ irrational.}
• Jul 5th 2011, 06:39 AM
Tinyboss
Re: Example of function not integrable and the square is
As girdav noted, it matters whether we're talking Riemann or Lebesgue. The rational/irrational function above is Lebesgue integrable. Think about the harmonic series to find a function not Lebesgue integrable but whose square is.
• Jul 6th 2011, 12:17 AM
CountingPenguins
Re: Example of function not integrable and the square is
Reimann
• Jul 6th 2011, 02:45 AM
girdav
Re: Example of function not integrable and the square is
Quote:

Originally Posted by CountingPenguins
Riemann

Hence Plato's answer solves the problem.