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Math Help - Question on proving the property of an integral

  1. #1
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    Question Question on proving the property of an integral

    Hi,

    I have a question which I'm having trouble with.

    Question on proving the property of an integral-meh.png

    I am told that I should first use the epsilon delta definition of continuity to prove that there must be a closed interval [c,d] with c < x0 < d such that for some x in [c, d], |h(x)| > |h(x0)|/2 by choosing epsilon to be |h(x0)|/2. I am having trouble with this.

    Any help would be appreciated
    Last edited by darkPassenger; May 24th 2012 at 08:50 AM.
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  2. #2
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    Re: Question on proving the property of an integral

    Just instantiate epsilon to |h(x0)| / 2 in the definition of continuity of h at x0. What do you get?
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  3. #3
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    Re: Question on proving the property of an integral

    Errr....well h is continuous at x0 when if |x - x0| < delta, then ||h(x)| - |h(x0)|| < |h(x0)|/2. Sorry but you might have to spell it out for me....
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  4. #4
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    Re: Question on proving the property of an integral

    Which, in turn, means that -h(x0)/2< h(x)- h(x0)< h(x0)/2. (You don't need absolute value on h(x0) because they are by hypothesis positive) and the absolute values on h(x) and h(x0) on the left are wrong.)

    Now add h(x0) to each part. h(x0)- h(x0)/2= h(x0)/2< h(x)< h(x0)/2+ h(x0)= 3h(x0)/2. And, again, by hypothesis h(x0) is positive.
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  5. #5
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    Re: Question on proving the property of an integral

    yep, made sense of it. Thanks for your help
    Last edited by darkPassenger; May 24th 2012 at 10:06 AM.
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