## henpen Group Title If $g(x)=\left\{\begin{matrix} \frac{2xy}{x^2+y^2} \text{ if }(x,y) \ne (0,0)\\ 0\text{ if }(x,y) = (0,0) \end{matrix}\right.$ Show that $g_y(0,0)$ and $g_x(0,0)$exist. one year ago one year ago

1. henpen Group Title

I'm not sure how to approach this- how do you find $g_y(x,y)$ in the first place if $g(x,y)$ is 2 functions glued together?

2. experimentX Group Title

that is just partial derivative, the function is extended function ...use the inequality to find the derivative.

3. experimentX Group Title

use g(x) = 2xy/(x+y)

4. henpen Group Title

Why would that work, given that at (0,0) the inequality does not hold necessarily?

5. experimentX Group Title

the function is continuous since f(0,0) = 0 and both limits are zero. show that the limit of partial derivatives f_x and f_y exits.

6. henpen Group Title

Is it always the case that if we apply a 'gluing job' to all discontinuities, the function will become continuous?

7. experimentX Group Title

yeah ... since this is removable discontinuity.

8. henpen Group Title

By the way, here is the limit at x=2 the same coming from both sides/|dw:1351706587261:dw|