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Continuity of multivariable functions

  1. Nov 1, 2013 #1
    1. The problem statement, all variables and given/known data

    A function f is defined on the whole of the xy-plane as follows:

    f(x,y) = 0 if x=0
    f(x,y) = 0 if y = 0
    f(x,y) = g(x,y)/(x^2 + y^2) otherwise

    a) g(x,y) = 5x^3sin(y)
    b) g(x,y) = 6x^3 + y^3
    c) g(x,y) = 8xy

    For each of the following functions g determine if the corresponding function f is continuous on the whole plane

    2. Relevant equations

    A function's limit exists if and only if it is not dependent of the path taken.

    3. The attempt at a solution

    Since the functions are continuous for all values of x and y, the only restriction on the xy plane is at the point (0,0). So I am trying to find the limit of these functions as (x,y) approaches (0,0).

    I have done so for c) using the line x=0 and y=x. These produce two different answers. Therefore, the limit of c does not exist at (0,0) and the function is not continuous on the xy plane.

    Any tips as to how I should tackle the other ones? I suspect that their limits are 0 (since every path I try gives 0), but I am having a hard time proving this.
     
  2. jcsd
  3. Nov 1, 2013 #2

    UltrafastPED

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    Science Advisor
    Gold Member

    Aren't they continuous at any point where the partial derivatives exist?
     
  4. Nov 3, 2013 #3
    Thanks. Turns out I misread the question. I ended up using your tip (partial derivatives) and was able to solve the problem.
     
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