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Math Help - Uniform Continuity Proof

  1. #1
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    Uniform Continuity Proof

    Hey guys,

    I'm having a little trouble trying to prove a question on uniform continuity that is in one of my textbooks (which doesn't have answers, unfortunately).

    The question asks:
    "Is the function f(x) = x3+x uniformly continuous on R? Prove your answer"

    I am almost certain that the answer is no for the simple reason that as x gets increasingly large the gaps get bigger so the same epsilon value won't work, but I'm not sure as to how to go about writing this in mathematical terms.

    I know that for a function to be uniformly continuous |f(x) - f(x')|<epsilon whenever |x - x'|<delta for ALL x, x' that are elements of the subset, S, which in this case is R.

    I would appreciate any help although preferably a proof that I can follow so that I can make the link to what I know to how to write it, if that makes sense.

    Thanks in advance,

    Mark
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Uniform Continuity Proof

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  3. #3
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    Re: Uniform Continuity Proof

    Thanks for this, it did help me with understanding it.
    Although I am now wondering how the '+ x' at the end would adjust this proof.

    I'm guessing that I would write "y3 + y - x3 - x",

    Which would then simplify to (y-x)(1 + (y2 + yx + x2)). Correct?

    But then when you pick values for x and y you cannot do so without there being a delta left over in your answer (ie., you can show the contradiction as there is a delta left over)

    Does this mean that it is, indeed, uniformly continuous?

    Or do I need to choose x and y so that y = x + d/(d+3) (d = delta)
    This would then cancel the deltas out and give 1 (as it is the multiplicative inverse), which contradicts the statement that |y2 + yx + x2| < 1.
    Although my problem with this is that I'm not sure if this fits with the previous restrictions that were made on x and y.

    Any help here would be great!

    Thanks,

    Mark
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