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I attempted some easy multiple-choice and T/F questinos to test if I am entirely clear about the topic before I do any proofy works. It's essential to be clear on these basic concepts and ideas. Let me know whether I am right or wrong and I'll appreciate if you briefly explain:) Thanks for helping! I want to check quickly if my answer is right.

You may just put the answers,or if you may, some explain would be great!

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  1. a. Yes. Because f(x) is always positive b. Yes. because of the term " x^2"

  2. a. no,because charactertistic function of the rational are not measure zero. therefore not Riemann integrable?? b. Probably no. Because again, charactertistic function of the rational are very "discrete".

3 a.Yes, Cantor set is measure zero. And one function is Riemann integrable as long as its discontinuity forms measure zero. b. Dounno. Honestly, I have no clue. Probably no again. where should I start?

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up vote 2 down vote accepted
  1. a) YES: $f$ strictly increases if $f'>0$ everywhere, b) NO: $f$ would be convex if $f''\ge 0$ was everywhere, but now $f''$ takes strictly negative as well.
  2. $\Bbb Q$ is Lebesgue measurable, has measure $0$ as a countable union of points. So, a) YES, b) YES only because that limit (the differential) would be $0$ anyway.
  3. a) YES, as you said, and b) NO, because now that limit would be $1$ for $[0,1]\setminus C$.
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