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!

Homework Help: Riemann Integral - little proof help

  1. Apr 14, 2009 #1
    1. The problem statement, all variables and given/known data

    Suppose f is integrable for all x in[a,b] and f(x)>C ( C is some constant),
    Must show that 1/f is also integrable.

    2. Relevant equations

    f is integrable implies Upf-Lpf<[tex]\epsilon[/tex] for some partition in [a,b]

    3. The attempt at a solution

    Therefore, I must come up with a good [tex]\epsilon[/tex] such that
    Lp(1/f) - Up(1/f) <[tex]\epsilon[/tex]

    Also f is bounded because it's integrable so there must be some m,M such that
    f([a,b])= [m,M]
    in other words f acheives it's minimum and maximum points on the interval.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Apr 16, 2009 #2
    Some hints. For simplicity let's just look at one subinterval [tex]I=[x_i,x_{i+1}][/tex]

    On I, we have [tex]m\le f(x)\le M[/tex] (although really there is no guarantee that m and M are achieved, we still know f is bounded as you said).

    Where does C fit in this inequality?

    Take reciprocals of the inequalities [tex]1/M\le 1/f(x)\le 1/m[/tex] and put 1/c in the correct place, and furthermore where is 0?

    Suppose M-m<delta.

    Then [tex]\frac1m-\frac1M=\frac{M-m}{Mm}[/tex] and you can find an upper bound for the last fraction in terms of delta and C.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook