# True/False Real Analysis Qualifying Exam Questions

The following is from a practice real analysis qualifying exam, and I had a couple questions about some of them.

$a$) I think this one is true, but I'm not sure how to prove it.

$b$) I know that $\partial^{\alpha}\hat{f} = \hat{[(-2\pi ix)^\alpha f]} (\xi)$. I don't necessarily see any reason that setting this equal to $0$ implies that $f=0$, so that would lead me to believe it might be false, but I'm unable to construct a counterexample.

$c$) We haven't done this yet, so skip.

$d$) True -- HW problem

$e$) Should be false if $d$ is true, but can't construct counterexample

$f$) I believe it's false and that $X=[0,1]$ with $f = \frac{1}{\sqrt x}\chi_{[0,1]}$ works.

$g$) I believe it's true using a comparison test argument