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Homework Help: Proof continuity of a function in R^3

  1. May 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Define F(x,y) = {1+x^2+y^2 when x>2^(1/2) AND y<2^(1/2)}
    {1-(x^2+y^2) when x>2^(1/2) OR y>2^(1/2)}

    Where in R^3 is F continuous? Prove it.

    2. Relevant equations

    definition of continuity

    3. The attempt at a solution
    I'm having a difficult time picturing how this function looks, but I think that F is continuous on R^3 except at (2^(1/2), 2^(1/2)).

    Let e>0. To show F is continuous at (a,b) I want to show there exists a d>0 such that |(x,y)-(a,b)|<d implies |f(x,y)-f(a,b)|<e. Let d=|(a,b)-(2^(1/2), 2^(1/2))|.
  2. jcsd
  3. May 29, 2010 #2


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    Staff Emeritus
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    Gold Member

    Going to epsilons and deltas is too much work. It helps to think about where the continuity condition can fail. If you're inside of one of the parts of the domain on which the function is piecewise defined, the function is obviously continuous just by looking at the definition of the function on that piece. So you're only interested in the boundary of those pieces. How can you decide (without doing too much work) whether the function is continuous on the boundary of those two parts of the domain?
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