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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.

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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?
that is just partial derivative, the function is extended function ...use the inequality to find the derivative.
use g(x) = 2xy/(x+y)

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Why would that work, given that at (0,0) the inequality does not hold necessarily?
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.
Is it always the case that if we apply a 'gluing job' to all discontinuities, the function will become continuous?
yeah ... since this is removable discontinuity.
By the way, here is the limit at x=2 the same coming from both sides/|dw:1351706587261:dw|

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