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Proving f(x) has a limit at all points not odd

  1. Oct 28, 2008 #1
    1.

    Define f:R→R as follows:
    f(x)= x- ⌊x⌋ if ⌊x⌋ is even.
    f(x)= x- ⌊x+1⌋ if ⌊x⌋ is odd.
    Determine those points where f has a limit and justify your conclusions (using δ and ε).



    2. Relevant equations



    3.

    The attempt at a solution involved a graph of the situations of f(x). With this graph my group and I determined f(x) has a limit at x_0 iff x_0 is not an odd integer. However, we hare having a hard time proving it, and, although the graph says we are right, we must use δ and ε in a formal proof.

     
  2. jcsd
  3. Oct 28, 2008 #2

    HallsofIvy

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    You have used some special symbols that do not show on my internet reader. What are they (describe in words)? Is it the "floor" symbol: [itex]\floor{x}[/itex]
     
  4. Oct 28, 2008 #3
    yes, it is the floor symbol, the others are delta and epsilon
     
  5. Oct 28, 2008 #4

    HallsofIvy

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    You have used special symbols that do not show on my internet reader. Is that the "floor function", f(x)= largest integer less than or equal to x? If so then the fact that floor(x+1)= floor(x)+ 1 should be helpful.
     
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