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Math Help - Continuity

  1. #1
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    Continuity

    I must say if the function is continuous in the point (0,0). Which is \displaystyle\lim_{(x,y) \to{(0,0)}}{f(x,y)}=f(0,0)

    The function:

    f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2}) & \mbox{ if }& y\neq{-x}\\1 & \mbox{if}& y=-x\end{matrix}

    I think its not continuous at any point, cause for any point I would ever have a disk of discontinuous points, but I must prove it. And I wanted to do so using limits, which I think is the only way to do it.

    \displaystyle\lim_{(x,y) \to{(0,0)}}{(x+y)^2\sin(\displaystyle\frac{\pi}{x^  2+y^2})}

    What should I do? should I use trajectories? the limit seems to exist, as the sin oscilates between -1 and 1, and the other part tends to zero.
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  2. #2
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    Quote Originally Posted by Ulysses View Post
    I must say if the function is continuous in the point (0,0). Which is \displaystyle\lim_{(x,y) \to{(0,0)}}{f(x,y)}=f(0,0)

    The function:

    f(x,y)=\begin{Bmatrix} (x+y)^2\sin(\displaystyle\frac{\pi}{x^2+y^2}) & \mbox{ if }& y\neq{-x}\\1 & \mbox{if}& y=-x\end{matrix}

    I think its not continuous at any point, cause for any point I would ever have a disk of discontinuous points, but I must prove it. And I wanted to do so using limits, which I think is the only way to do it.

    \displaystyle\lim_{(x,y) \to{(0,0)}}{(x+y)^2\sin(\displaystyle\frac{\pi}{x^  2+y^2})}

    What should I do? should I use trajectories? the limit seems to exist, as the sin oscilates between -1 and 1, and the other part tends to zero.

    \lim\limits_{(x,y)\to (0,0)} (x+y)^2=0\,\,\,and\,\,\,\sin\left(\frac{\pi}{x^2+y  ^2}\right) is bounded, so the limit does exists and equals..., and thus the function isn't continuous at the origen.

    Tonio
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