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Math Help - Uniform Continuity

  1. #1
    Senior Member slevvio's Avatar
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    Uniform Continuity

    Hello there. I am supposed to show whether or not the function

    0,1) \rightarrow \mathbb{R} " alt=" f0,1) \rightarrow \mathbb{R} " /> given by  f(x) = \frac{1}{x^2}

    is uniformly continuous.

    I suspect it is not so I'm trying to prove the statement:

     \exists \epsilon > 0 such that  \forall \delta > 0 \exists x,a \in (0,1) such that  |x-a| < \delta and  | \frac{1}{x^2} - \frac{1}{a^2} | < \epsilon .

    i.e. our choice of delta in the definition of continuity does not depend on a. Any help with this would be appreciated.
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  2. #2
    Senior Member slevvio's Avatar
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    Addendum:

    If we have a function f continuous on [a,b] is it true that f is uniformly continuous on (a,b) assuming f(a) and f(b) are defined? If so why?

    What about if f(a) is not defined? For example f(x) = 1/(x^2) is continuous on [0,1] (and so is uniformly continuous on [0,1]) but is funiformly continuous on (0,1) considering f(0) is not defined? If so why?
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  3. #3
    Super Member
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    Quote Originally Posted by slevvio View Post
    Addendum:

    If we have a function f continuous on [a,b] is it true that f is uniformly continuous on (a,b) assuming f(a) and f(b) are defined? If so why?
    Yes, remember that uniform continuity is a set property and it's trivial to show from the definition that if f is unif. cont. on A then it is so on every B \subseteq A

    Quote Originally Posted by slevvio View Post
    What about if f(a) is not defined? For example f(x) = 1/(x^2) is continuous on [0,1] (and so is uniformly continuous on [0,1]) but is funiformly continuous on (0,1) considering f(0) is not defined? If so why?
    How is f going to be unif. cont. on [a,b] when it's not even defined in a?
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  4. #4
    Senior Member slevvio's Avatar
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    thanks a very helpful post
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  5. #5
    Senior Member slevvio's Avatar
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    how can I show that f in my original question is not uniformly continuous though? since it is actually continuous on (0,1) ?
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  6. #6
    Math Engineering Student
    Krizalid's Avatar
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    http://home.iitk.ac.in/~psraj/mth101...es/uniform.pdf

    on page 3 there's an useful fact.
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