Let
.
By using the definition of differentiability show that f is not differentiable at (0,0).
Definition is:
I already know that . Also the increment is . So
Now, I need to evaluate this to show that it doesn't =0. I'm stuck here. I don't know if I should use the conjugate OR use polar coordinates to evaluate this limit? Can someone help?
Yes but then how are we going to evaluate this limit?
I mean if
And in the expression on the RHS there are no more terms left, and we don't know the value of m! So, what do I need to do?
Of course, I could rationalize the denominator but I don't know how to simplify the numerator in the following:
You actually do know the value of m - you can set it to be anything you like!
That is why the function is not differentiable around (0,0) - because the limit depends on m (ie. you can get two different values for the limit for two different values of m) and as a result, it does not exist (do you understand why?)