## henpen 3 years ago 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.

1. henpen

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

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

3. experimentX

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

4. henpen

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

5. experimentX

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

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

7. experimentX

yeah ... since this is removable discontinuity.

8. henpen

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