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Math Help - Determine where f is continuous and where f is discontinuous

  1. #1
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    Determine where f is continuous and where f is discontinuous

    I have never liked problems of the sort where rational/irrational numbers are involved.

    For the function f(x,y)=\left\{\begin{array}{cc}x^2+y^2,&x,y\mbox{ both rational }\\0, & \mbox{ otherwise, }\end{array}\right
    determine where f is continuous and where f is discontinuous. Does f have any partial derivatives?

    A note provided is as follows: Every neighborhood in R^2 always contains points with rational and points with irrational coordinates.

    Mainly, the part I have difficulty with is the rational/irrational bit. I've never been able to work with those properly, probably because I missed something fundamental in my earlier years.
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  2. #2
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    Quote Originally Posted by Runty View Post
    I have never liked problems of the sort where rational/irrational numbers are involved.

    For the function f(x,y)=\left\{\begin{array}{cc}x^2+y^2,&x,y\mbox{ both rational }\\0, & \mbox{ otherwise, }\end{array}\right
    determine where f is continuous and where f is discontinuous. Does f have any partial derivatives?

    A note provided is as follows: Every neighborhood in R^2 always contains points with rational and points with irrational coordinates.

    Mainly, the part I have difficulty with is the rational/irrational bit. I've never been able to work with those properly, probably because I missed something fundamental in my earlier years.
    note that by the squeeze theorem that

    0 < f(x,y)<x^2+y^2 \forall (x,y) \in \mathbb{R}^2

    Now just let \epsilon=\delta and when

    |(x,y)-(0,0)|< \delta \iff x^2+y^2<\delta

    then |f(x,y)-0|\le |x^2+y^2|=...

    Will this work for any other point than (0,0)?
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