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Show that arctan(x) exists and is continuous

  1. Feb 7, 2010 #1
    1. The problem statement, all variables and given/known data

    Show tan(x): (-pi/2, pi/2) -> R has a continuous inverse arctan(x) : R -> (-pi/2, pi/2).

    You may assume that tan(x) is continuous and strictly increasing on the given domain, and tends to +/- [tex]\infty[/tex] at +/- pi/2

    2. Relevant equations



    3. The attempt at a solution

    I think I have shown that tan(x) has an inverse on this domain by showing it is bijective.

    However, I am unsure how to go about showing that arctan(x) is continuous.
     
  2. jcsd
  3. Feb 7, 2010 #2

    vela

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    You could try considering tan(arctan(x))=x and show that if arctan(x) is not continuous, it leads to a contradiction.

    (I'm just offering a suggestion. I think it'll work, but I didn't actually try it.)
     
  4. Feb 7, 2010 #3

    Hurkyl

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    Continuous functions can have discontinuous inverses.

    I'm not sure if this is possible when the domain and codomain are intervals, though. I suspect not.
     
  5. Feb 7, 2010 #4
    What characterizations of continuity have you learnt? Try showing that tan takes every open set to an open set.
     
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