Example of a non-integrable function f , such that |f| and f^2 are integrable?

[irresistible]

I'm looking for a function
f:[a,b] -> R such that |f| and f² are integrable on [a,b]
any helps?

 

[adriank]

To start with, f must be discontinuous (a continuous function on a closed interval is integrable). Can you think of a function f such that f is discontinuous, but |f| (and thus f² = |f|²) is continuous?This is really an analysis question, so it really belongs in that forum. :)[/size]

 

[LuisVela]

f(x)= 1 for x rational,
-1 for x irational

 

[adriank]

Don't just give him an answer. ;)

(That's the exact same function I was thinking of, though.)

 

[LuisVela]

I forgot to say: Only if you are dealing with Newton or rieman integrals, otherwise f(x) is integrable.

 

[irresistible]

Oh I don't know why I didn't think of that!
Thank you so much to both of youuuu!