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Math Help - I think I am making this harder

  1. #1
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    Unhappy I think I am making this harder

    the definition of continutiy is stated in garbled forms. Identify the one that is equivalent for the definition of continuity and draw a sketch of what each of the others represents.

    a. for every epsilon > 0 and every delta > 0, |x - c| < delta implies |f(x) - f(c)| < epsilon.

    b. There is an epsilon >0 such that for every delta>0, |x - c| <delta implies
    |f(x) - f(c)| < epsilon.

    c. for some epsilon > 0, there is a delta > 0 such that |x - c| < delta implies |f(x) - f(c)| < epsilon.

    d. There is a delta >0 such that for every epsilon >0, |x - c| <delta implies
    |f(x) - f(c)| < epsilon.


    I chose d. I don't know for sure.
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  2. #2
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    Quote Originally Posted by Susie38
    the definition of continutiy is stated in garbled forms. Identify the one that is equivalent for the definition of continuity and draw a sketch of what each of the others represents.

    a. for every epsilon > 0 and every delta > 0, |x - c| < delta implies |f(x) - f(c)| < epsilon.

    b. There is an epsilon >0 such that for every delta>0, |x - c| <delta implies
    |f(x) - f(c)| < epsilon.

    c. for some epsilon > 0, there is a delta > 0 such that |x - c| < delta implies |f(x) - f(c)| < epsilon.

    d. There is a delta >0 such that for every epsilon >0, |x - c| <delta implies
    |f(x) - f(c)| < epsilon.


    I chose d. I don't know for sure.
    THe definition for coutinouity of real function is that,
    \lim_{x\to c}f(x)=f(c)
    By definition we have,
    \forall \epsilon>0 there is a \delta>0 such as, |f(x)-f(c)|<\epsilon whenever |x-c|<\delta.

    The meaning of "whenever" is the same as this implies something else. Thus it is a saying,
    .... |x-c|<\delta implies |f(x)-f(c)|<\epsilon thus (d) is the correct choice.
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  3. #3
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    How do I use a graph to represent the other 3 choices that were not the right answer?
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