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August 5th, 2017, 06:29 AM  #1  
Newbie Joined: Jun 2017 From: Earth Posts: 16 Thanks: 0  A question about a sufficient condition of differentiability of a multivariable func
I see many times a sufficient condition of differentiability of a multivariable function: Quote:
$\displaystyle f(x,y) = \left\{ \begin{array}{*{20}{c,l}} \frac{x^2y}{x^2 + y^2}&{\rm{, if}} (x,y) \ne (0,0)\\ 0 & {\rm{, if}} (x,y) = (0,0) \end{array} \right.$. It has constant value 0 along x and yaxis, thus constant and therefore continuous partial derivative 0 along x and yaxis, but it is not differentiable at (0,0) because the limit $\displaystyle \lim \limits_{\Delta x \to 0,\Delta y \to 0} \frac{[f(\Delta x,\Delta y)  f(0,0)][f_x'(0,0)\Delta x + f_y'(0,0)\Delta y]}{\rho}$ where $\displaystyle \rho=\sqrt{\Delta x^2+\Delta y^2}$ does not exist (approaching along lines of different slopes). So where is the problem? Is the sufficient condition mentioned above wrong or did I miss anything when applying it? Thank you.  
August 5th, 2017, 10:11 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,576 Thanks: 667 
I don't know where you saw your quote: Quote:
A function f(x,y) is differentiable at $\displaystyle (x_0, y_0)$ if both its partial derivatives are continuous in some neighborhood of that point. Last edited by skipjack; August 5th, 2017 at 05:00 PM.  
August 5th, 2017, 11:34 AM  #3 
Newbie Joined: Jun 2017 From: Earth Posts: 16 Thanks: 0 
What?

August 6th, 2017, 03:19 AM  #4  
Newbie Joined: Jun 2017 From: Earth Posts: 16 Thanks: 0  Quote:
 
August 6th, 2017, 03:35 AM  #5  
Senior Member Joined: Oct 2009 Posts: 141 Thanks: 59  Quote:
 

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condition, differentiability, func, multivariable, question, sufficient 
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