An example of a function f that is not Riemann integrable but |f| is Riemann integrable

I'm trying to think of an example where this happens so that I can practice showing that a function is or is not Riemann integrable, and thought this would be an interesting case to try. But I cannot work out when this would happen. Any ideas? Thank you :)

Here's a hint: your function needs to switch between being positive and negative, a lot.
Stronger hint: the rationals are a somewhat large set.





Answer: Let f:[0, 1]->R be 1 on irrationals, -1 on rationals. The lower Riemann sum is -1, the upper is 1, so it's not Riemann integrable. |f| is just 1, so is obviously Riemann integrable. This is a wonderful motivating example for higher forms of integration, like the Lebesgue integral. The Lebesgue integral assigns 1 to the integral of f, which makes pretty solid intuitive sense since the irrationals are uncountable and the rationals are countable.