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Riemann integration

  1. May 9, 2010 #1
    1. The problem statement, all variables and given/known data

    If a<c<d<b and f is integrable on (a,b), show that f is integrable on (c,d)

    2. Relevant equations

    3. The attempt at a solution

    I know that f is integrable on (a,b) iff for all e>0 there exists step functions g and h such that g [tex]\leq[/tex] f1(a,b) [tex]\leq[/tex] h and I(g-h) <e
    ( 1(a,b) in the indicator function and I(g-h) is the integral of the step functions)

    I feel like this should allow me to fairly easily show that f is also integrable on (c,d) but I just don't know how to start.

    Do I need to consider partitions?

  2. jcsd
  3. May 9, 2010 #2


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    I assume you mean [itex]I(h-g)[/itex], not [itex]I(g-h)[/itex].

    To show integrability on the interval [itex](c,d)[/itex], consider the functions [itex]g|_{(c,d)}[/itex] and [itex]h|_{(c,d)}[/itex], which are the restrictions of [itex]g[/itex] and [itex]h[/itex] to the interval [itex](c,d)[/itex]. Are the restrictions still step functions? Do they satisfy the desired inequality?
  4. May 9, 2010 #3
    Yes I did sorry.
    Thanks :) so if I use those functions that take the same value on (c,d) and are 0 elsewhere I think I can see how it goes.
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