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Math Help - Lower and Upper Darboux Sums

  1. #1
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    Lower and Upper Darboux Sums

    Let f(x) : [a,b]  -> R be a bounded function such that U(f,P) = L(f,P) for any partition P of [a,b].
    Prove that there exists a constant c such that f(x) = c for any x in [a,b] (by contradiction) .
    Appreciate if someone could help on this one. Thanks...
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  2. #2
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    Quote Originally Posted by seniorcalculus View Post
    Let f(x) : [a,b] -> R be a bounded function such that U(f,P) = L(f,P) for any partition P of [a,b].
    Prove that there exists a constant c such that f(x) = c for any x in [a,b] (by contradiction) .
    Appreciate if someone could help on this one. Thanks...
    it's really trivial! let P=\{a,b\}. let M=\sup_{x \in [a,b]}f(x), \ m=\inf_{x \in [a,b]}f(x). then: 0=U(f,P)-L(f,P)=(M-m)(b-a). thus M=m, which means f is constant.
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  3. #3
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    Okay, thanks. But that's a direct proof, right? I kinda need help for proof by contradiction though.
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  4. #4
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    Quote Originally Posted by seniorcalculus View Post
    Okay, thanks. But that's a direct proof, right? I kinda need help for proof by contradiction though.
    it's the same proof! let P,M, m be as above. suppose f is not constant. then M \neq m. but 0=U(f,P)-L(f,P)=(M-m)(b-a). thus b=a. contradiction!
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  5. #5
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    Okay, I understand it now. Thanks, NonCommAlg
    But, I realized that you have picked an instance of a partition in this case, which is P={a,b}. But in the proof we have to show that for any partition interval. Shouldn't we have to assume for any partition P?
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