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For what domains is this function continuous for?

  1. Mar 30, 2008 #1
    1. The problem statement, all variables and given/known data

    f(x,y) = 1/(x^2 + y^2 -1)

    1. For what domains is f continuous?
    2. For what domains is f a C^1 function? (Here C^1 means that the first derivatives of f are all continuous)


    2. Relevant equations



    3. The attempt at a solution

    I would be very grateful for the help you can give me with this, because I'm just not sure if I'm coming to the right conclusions or if there is something I'm missing. If you could also keep to relatively simple maths as I haven't done a lot before (keeping to simple explanations or methods would be so helpful).

    For the first part I believe that f would be continuous when the denominator is not zero. So what I get for the first part is that it is continuous for all domains apart from:
    (x , + or - SQRT(1 - x^2)) where x is any real number

    For the second part I get the following two first derivatives:
    df/dx = (-2x)/((x^2 + y^2 - 1)^2)
    df/dy = (-2y)/((x^2 + y^2 - 1)^2)
    And from this I get that f is C^1 for:
    (x , + or - SQRT(1 - x^2)) where x is any real number

    Can anyone give me some help with this? I'm not sure if I am on the right track with my answers or if I have missed some domains for which it is not or is continuous/C^1 for.

    Thank you for helping,

    Jenny
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 30, 2008 #2

    benorin

    User Avatar
    Homework Helper

    You got it. You may want to give the domain as {(x,y) | x^2+y^2[itex]\neq[/itex]1 }.
     
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