
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 5th, 2017, 07:29 AM  #1  
Newbie Joined: Jun 2017 From: Earth Posts: 17 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, 11:11 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 2,944 Thanks: 797 
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 06:00 PM.  
August 5th, 2017, 12:34 PM  #3 
Newbie Joined: Jun 2017 From: Earth Posts: 17 Thanks: 0 
What?

August 6th, 2017, 04:19 AM  #4  
Newbie Joined: Jun 2017 From: Earth Posts: 17 Thanks: 0  Quote:
 
August 6th, 2017, 04:35 AM  #5  
Senior Member Joined: Oct 2009 Posts: 232 Thanks: 82  Quote:
 

Tags 
condition, differentiability, func, multivariable, question, sufficient 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Necessary vs Sufficient condition  matisolla  Algebra  1  May 26th, 2015 04:38 AM 
necessary & sufficient condition of utility maximization  helloprajna  Economics  4  August 30th, 2013 04:46 AM 
simple 2sinx func question  peaceofmind  Calculus  2  May 11th, 2011 10:40 AM 
Differentiability and Lipschitz condition on an interval og  Artur  Real Analysis  1  June 27th, 2009 03:10 AM 
A question of Delta in definition of Limit of Square func...  Amir Kiani  Calculus  2  January 26th, 2009 01:17 AM 