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Math Help - Real Analysis: continuity

  1. #1
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    Real Analysis: continuity

    Sorry, only a few more, lasts ones I'm not sure of, final on Friday, these are some of the review problems I don't get:

    1) Show 2^x = 3x for some xE (0,1)

    2) let f(x) = { x xrational
    0 xirrational

    a. prove lim f(x) = 0
    x->0
    b) Prove that lim f(x), does not exist for c not equal to 0.
    x--> c

    3) Suppose f is uniformly continuous on [a,b] and uniformly continuous on [b,c] Prove using an epsilon-delta argument that f is uniformly continuous on [a,c]

    Thanks, this professor does a horrible job at teaching, and I'm really lost, any help, even how to start any of these would be great, thanks!
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  2. #2
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    Quote Originally Posted by Vitava61 View Post
    Sorry, only a few more, lasts ones I'm not sure of, final on Friday, these are some of the review problems I don't get:

    1) Show 2^x = 3x for some xE (0,1)

    2) let f(x) = { x xrational
    0 xirrational

    a. prove lim f(x) = 0
    x->0
    b) Prove that lim f(x), does not exist for c not equal to 0.
    x--> c

    3) Suppose f is uniformly continuous on [a,b] and uniformly continuous on [b,c] Prove using an epsilon-delta argument that f is uniformly continuous on [a,c]

    Thanks, this professor does a horrible job at teaching, and I'm really lost, any help, even how to start any of these would be great, thanks!

    for 1)
    consider the function f(x)=2^x-3x
    Hint: use the Intermediate Value Threom

    for 2) How would you prove

    \lim_{x \to 0}f(x)=0 if f(x)=x

    compare with the above problem and adapt the proof.

    for 3)
    uniform continity state that for every epslion > 0 there exits a delta

    fix epsilon > 0 find a delta on [a,b] and [b,c]

    How could we get a delta that would work on [a,c]?

    I hope this helps.

    Brett
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