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Math Help - another darboux question

  1. #1
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    Wink another darboux question

    Let K be a D-domain and let f:K--R be differentiable on the interval [a,b] as a subset of K (a< b).


    f(x) = { 1 if x>0
    { -1 if x<0
    is the derivate of f(x) = |x|.
    Why doesn't this contradict Darboux's theorem?

    I don't understand how this DOESN'T contradict Darboux. There is discontinuity in the f'(x) graph because it jumps from -1 to 1. Also it violates the IVT because there is no u for which f(a)<u<f(b).

    Please help
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  2. #2
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    Quote Originally Posted by derek walcott View Post
    Let K be a D-domain and let f:K--R be differentiable on the interval [a,b] as a subset of K (a< b).


    f(x) = { 1 if x>0
    { -1 if x<0
    is the derivate of f(x) = |x|.
    Why doesn't this contradict Darboux's theorem?

    I don't understand how this DOESN'T contradict Darboux. There is discontinuity in the f'(x) graph because it jumps from -1 to 1. Also it violates the IVT because there is no u for which f(a)<u<f(b).

    Please help
    f(x)=|x| isn't derivable at x=0 and thus there's no contradiction to anything.

    Tonio
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by derek walcott View Post
    Let K be a D-domain and let f:K--R be differentiable on the interval [a,b] as a subset of K (a< b).


    f(x) = { 1 if x>0
    { -1 if x<0
    is the derivate of f(x) = |x|.
    Why doesn't this contradict Darboux's theorem?

    I don't understand how this DOESN'T contradict Darboux. There is discontinuity in the f'(x) graph because it jumps from -1 to 1. Also it violates the IVT because there is no u for which f(a)<u<f(b).

    Please help
    As tonio said Darboux's theorem says that if f is continuous on [a,b] and differentiable on (a,b) then for any \min\left\{f'_{-}(a),f'_{+}(b)\right\}<k<\max\left\{f'_{-}(a),f'_{+}(b)\right\} there exists some c\in(a,b) such that f'(c)=k. Thus, Darboux's theorem is not violated here since f is not differentiable on ANY open ball of 0. So there is no way to even apply it.
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