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  1. #1
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    Suppose  f(x) = \begin{cases} 0 \ \ \ \text{if} \ \ x \notin \mathbb{Q} \\ 1/q \ \ \ \text{if} \ x \in \mathbb{Q} \ \text{and} \ x = p/q \ \text{in lowest terms} \end{cases}

    1. Fix  t \in \mathbb{R} and  n \in \mathbb{N} . Show that  f maps only finitely many elements of  (t-1/2, t+1/2) to  1/n .

    So the length of this interval is  1 . I think its  n-1 . But that is intuition. How do you make it rigorous?


    2. Prove that  f is continuous at every irrational number. So  \forall \epsilon >0, \exists \delta >0 such that  |x-a| < \delta \implies |f(x)| < \epsilon . Do we want to elements that dont map to  1/n ?
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  2. #2
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    Basically we are trying to show that  f maps a finite number of rational numbers in lowest terms in the interval  I =  (t- \frac{1}{2}, t+\frac{1}{2}) to  \frac{1}{n} . Let  x =  p/n \in I and in lowest terms. Then  p and  n are coprime. Can we somehow use this to deduce that the maximum number is  n-1 ? number of fractions  p/n in interval is  n . number in lowest terms is  \leq n-1 .
    Last edited by manjohn12; April 17th 2009 at 08:13 PM.
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