1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Riemann Integrability of Composition

  1. Apr 11, 2013 #1
    1. The problem statement, all variables and given/known data
    Let ψ(x) = x sin 1/x for 0 < x ≤ 1 and ψ(0) = 0.
    (a) If f : [-1,1] → ℝ is Riemann integrable, prove that f [itex]\circ[/itex] ψ is Riemann integrable.
    (b) What happens for ψ*(x) = √x sin 1/x?

    2. Relevant equations
    I've proven that if ψ : [c,d] → [a,b] is continuous and for every set of measure zero Z [itex] \subset [/itex] [a,b], [itex] ψ^{\text{pre}}(Z) [/itex] is a set of measure zero in [c,d], then if f is Riemann integrable, f [itex]\circ[/itex] ψ is Riemann integrable. However, this doesn't apply well in this situation. What can I do? I have a hunch that both f [itex] \circ [/itex] ψ and f [itex] \circ [/itex] ψ* are Riemann integrable.

    3. The attempt at a solution
  2. jcsd
  3. Apr 11, 2013 #2
    Any suggestions?
  4. Apr 11, 2013 #3
    What do you know about Riemann integration?? What theorems could come in handy?
    For example, do you know that a function is Riemann integrable iff the set of discontinuities has measure 0?
  5. Apr 11, 2013 #4
    SammyS: My apologies, it was not twenty-four hours.

    Micromass: I am aware of that theorem, and in fact used it to prove the theorem in the original post. The problem is that I do not know how to show that the set of discontinuities of [itex] f \circ \psi [/itex] are of measure 0. Clearly it is discontinuous at a point u iff f is discontinuous at [itex] \psi(u) [/itex], but I don't know where to go from here.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted

Similar Discussions: Riemann Integrability of Composition
  1. Riemann integration (Replies: 2)

  2. Riemann integrability (Replies: 1)

  3. Riemann integrability (Replies: 1)

  4. Riemann Integrability (Replies: 2)