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Example in Spivak's Calculus

  1. Dec 31, 2016 #1
    1. The problem statement, all variables and given/known data

    Consider the function
    $$
    f(x) := \begin{cases}
    0, & if~ x \in (0,1) - \mathbb{Q} \\
    \frac{1}{q}, & if~ x = \frac{p}{q} \in (0,1) \cap \mathbb{Q} \mbox{ in lowest terms. } \\
    \end{cases}
    $$

    In his Calculus book, Spivak shows us that ##\lim_{x \rightarrow a} f(x) = 0## for all ##a \in (0,1)##. I am having trouble with a few of his remarks
    2. Relevant equations


    3. The attempt at a solution

    First, he "let ##n## be a natural number so large that ##\frac{1}{n} \le \epsilon##." This is fine, I suppose; this is just the Archimedean property, although I have always taken ##\frac{1}{n}## strictly less than ##\epsilon##. He then goes on to say "that the only numbers ##x## for which ##|f(x)-0| < \epsilon## could be false are:

    $$\frac{1}{n},....,\frac{n-1}{n}$$"

    I find this rather perplexing...Why wouldn't ##x = \frac{1}{n}## satisfy ##|f(x)| < \epsilon##, for example? Spivak's aim seems to be choosing a ##\delta## such that any ##x## in ##(a-\delta, a+\delta)## is not any one of the numbers in the above list but is smaller, so that ##|f(x)| < \epsilon##. But, as I ask, why wouldn't ##x = \frac{1}{n}## work?

    PS I can't figure out why there are horizontal lines through my text.
    Mod note: Fixed this. For some reason the overstruck text was surrounded by overstrike BB code tags -- [ s] and [ /s]. I removed them.
     
    Last edited by a moderator: Dec 31, 2016
  2. jcsd
  3. Dec 31, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I had such horizontal lines in one of my posts about two weeks ago. The only cure was to log off and re-boot my computer. It worked for me, maybe not for you.
     
  4. Dec 31, 2016 #3

    Mark44

    Staff: Mentor

    See my added note in the first post of this thread.
     
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