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Thread: f(x) continuous at c

  1. #1
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    f(x) continuous at c

    Please Help!!

    Prove that if c is irrational f(x) is continuous at c.

    I would appreciate any type of hints or just where to start.
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  2. #2
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    Re: f(x) continuous at c

    think it would help if you posted what f(x) is ... ?
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    Re: f(x) continuous at c

    Quote Originally Posted by cloeannx3 View Post
    Please Help!!
    Prove that if c is irrational f(x) is continuous at c.
    I would appreciate any type of hints or just where to start.
    How can we help you? We are blind here.

    You told us absolutely nothing about $\displaystyle f$!
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  4. #4
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    Re: f(x) continuous at c

    I'm so sorry. I didn't even realize that I left that part out. (It's been a long week)

    f(x) is defined on the interval [0,1] to be 0 if x xis irrational and to be 1/n if x is rational and if x can be written fully reduced as x =m/n
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  5. #5
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    Re: f(x) continuous at c

    suppose ε > 0 is given. we need to find some δ > 0 so that |x-c| < δ means |f(x)| < ε.

    well pick a natural number n such that 1/n < ε. then there are only finitely many points (all of them rational) in the interval [0,1] with f(x) ≥ 1/n (let's call them x1,...,xk).

    let δ = min(|x1 - c|, |x2 - c|,...,|xk - c|). does that work?

    (for example, if ε = 0.4, we could use n = 3. the only points for which f(x) ≥ 1/3 are {0,1/3,1/2,2/3,1}, pick the one that c is closest to (for example if c = √2 - 1, we would pick δ = 4/3 - √2, since 1/3 is the closest point)).
    Last edited by Deveno; Dec 4th 2012 at 04:52 PM.
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