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Math Help - Continuity of a two variable function

  1. #1
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    Continuity of a two variable function

    The question is :

    Prove that f is continuous at the point (0,0) where f is

    f(x,y)= ysin(1/x) , when x is not equal to zero
    0 , when x=0.

    I have to prove this using the epsilon-delta definition of limit. Can't figure out how to do that ?

    Can somebody please explain the procedure .
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  2. #2
    MHF Contributor

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    Re: Continuity of a two variable function

    Do you not know what the "epsilon-delta definition of a limit? For a function of two variables, you must show
    "For any \epsilon> 0 there exist \delta> 0 such that if \sqrt{x^2+ y^2}< \delta then |f(x, y)- L< \epsilon[/tex].

    Here, your function is y sin(1/x) and you want the limit to be the value of the function, 0. So you want to get |y sin(1/x)|< \epsilon for x and y sufficiently small. I would use the fact that |y sin(1/x)|= |y||sin(x)|\le |y| because -1\le sin(1/x)\le 1 for any x not equal to 0.
    Thanks from mrmaaza123
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