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Math Help - Continuity of a function of 2 variables

  1. #1
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    Continuity of a function of 2 variables

    Hi, can someone just tell me if my argument for this question is correct (and if it isn't, what is the correct answer)

    Is the function f : R^2 -> R defined as f(x,y) = {(sin xy)/x for nonzero x
    y for x=0 }

    continuous at (0,0) ?

    My answer: If we take any two sequences of real numbers a_n and b_n that tend to zero as n to infinity, then we have that (a_n, b_n) -> (0,0) and f(a_n, b_n) = sin(a_n b_n)/a_n and since for small values of x, sin(x) is very close to x we have that f(a_n, b_n) has the same limit as a_n b_n / a_n = b_n which goes to 0 = f(0,0). Thus it is continuous at (0,0).

    Thanks
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  2. #2
    Super Member General's Avatar
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    Your function split into two functions at x=0 only .. or at the point (0,0) .. ?!!
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  3. #3
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    Quote Originally Posted by General View Post
    Your function split into two functions at x=0 only .. or at the point (0,0) .. ?!!
    What do you mean? It is (sin xy)/x when x is not 0, and y when x is 0. So for example f(0,3) = 3 ... f(0,0) = 0 ..... f(2,3) = (sin 6)/2
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