- #1

Lee33

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## Homework Statement

Where is the function ##f:E^2\to\mathbb{R}## given by ##f(x,y)=\begin{cases}\frac{xy}{|x|+|y|} & , \ \text{if} \ (x,y)\ne(0,0)\\

0 & , \ \text{if} \ (x,y)=(0,0) \end{cases}## differentiable?

## Homework Equations

None

## The Attempt at a Solution

The function is continuous so the partials exists, thus we have ##\frac{\partial f}{\partial x} = \begin{cases} \frac{y(|x|+|y|)\pm xy|y|}{(|x|+|y|)^2}, & x>0\\

\frac{y(|x|+|y|)\pm xy|y|}{(|x|+|y|)^2}, & x<0. \end{cases}##

##\frac{\partial f}{\partial y} = \begin{cases} \frac{x(|x|+|y|)\pm xy|x|}{(|x|+|y|)^2}, & y>0\\

\frac{x(|x|+|y|)\pm xy|x|}{(|x|+|y|)^2}, & y<0. \end{cases}##

For ##(x,y)\ne (0,0),## the partials ##\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}## exists and are continuous so ##F## is differentiable at any ##(x,y)\ne (0,0). ##

For ##(x,y)=(0,0),## I am having trouble showing why the partials are not continuous at ##(0,0).##