1. ## Riemann integrable

Let
f : [a, b]--->R be bounded. Prove or disprove: if $f^{2}$ is Riemann integrable on [a, b] then f is Riemann integrable on [a, b].

I have a set of theorems in my notes but I can't find the correct one to apply to the above problem.

if $f^{2}$ is Riemann integrable on [a, b] then f is has also Riemann integrable on [a,b] correct? How do i prove this?

thanks for any help.

2. For the first one, without knowing what theorems you have available, I don't see how we can assist. I would have used "a bounded function is integrable on [a, b] if and only if its set of points of discontinuity in [a, b] has measure 0". Do you have that?

As for the second problem, consider f(x)= 1 if x is rational, -1 if x is irrational.

3. Originally Posted by HallsofIvy
For the first one, without knowing what theorems you have available, I don't see how we can assist. I would have used "a bounded function is integrable on [a, b] if and only if its set of points of discontinuity in [a, b] has measure 0". Do you have that?

As for the second problem, consider f(x)= 1 if x is rational, -1 if x is irrational.
Aren't the first and second questions the same?

4. I have only question, the first question.

I just wrote it the other way round in the second part.

thanks