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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

    micromass

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    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.
     
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