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Math Help - condition for differentiability

  1. #1
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    condition for differentiability

    I have to determine if f(x,y) = \frac{x}{y} + \frac{y}{x} is differentiable at all points in its domain and of class C^1

    \frac{\partial f}{\partial x} = \frac{1}{y} - \frac{y}{x^2}
    \frac{\partial f}{\partial y} = \frac{1}{x} - \frac{x}{y^2}

    seems that if (x,y)=(0,0) then its not continuous or differentiable

    and if x=0 or y=0 then one or both partials fail to be continuous or differentiable

    so no, its not of class C^1

    correct?
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  2. #2
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    Re: condition for differentiability

    $C^1$ is the class of functions who's first derivatives are continuous at all points their domain.

    $(0,0)$ is not in the domain of $f(x,y)$

    Are there any points in the domain of $f(x,y)$ where the first partials are discontinuous?
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  3. #3
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    Re: condition for differentiability

    Quote Originally Posted by romsek View Post
    $C^1$ is the class of functions who's first derivatives are continuous at all points their domain.

    $(0,0)$ is not in the domain of $f(x,y)$

    Are there any points in the domain of $f(x,y)$ where the first partials are discontinuous?
    if (0,0) is not in the domain then theres no points where the partials are discontinuous so it would be C^1. and (0,0) is not in it because it is not defined there, duh haha.
    Last edited by Jonroberts74; June 28th 2014 at 12:27 PM.
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  4. #4
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    Re: condition for differentiability

    More to the point: none of the points (x, y) with either x or y zero, are in the domain.
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