Define this Function on [0,1]

Question

I need a function, $f$ that is bounded but not integrable on [0,1], but $f^2$. I am thinking I should modify the Dirichlet function and use it to create a function whose square is constant. I am stuck and cannot think of it.

Answer 1

In fact, a variation of the Dirichlet function serves. Let $f:[0,1]\rightarrow \mathbb{R}$ given by $f(x)=1 $ if $x$ is rational and $f(x)=-1$ if $x$ is not rational. Then, $f$ is bounded, not integrable and $f^2\equiv 1$, an so integrable.

Answer 2

Using the Dirichlet function is a good idea. Consider one that is $-1$ for rationals and $1$ for irrationals.