Continuous and differentiable points of f(x,y,z) (Existence of multivariable limit?)
The problem is stated as follows:
Find the continuous points P and the differentiable points Q of the function in , defined as
If you want to look at the limit I'm having trouble with, just skip a few paragraphs to the sentence that begins with a red word. I'm mostly including the rest in case anyone is in the mood to point out flaws in my reasoning.
Differentiating with respect to x, y and z, respectively (when will make it apparent that all three partials will contain a denominator of and a continuous numerator. Thus, these partials are continuous everywhere except in , and it follows that is differentiable, and consequently, also continuous in all points .
Investigating if is differentiable at , we investigate the limit
Evaluating along the line , that is, , it is found after a bit of work and one application of l'H˘pital's rule that the limit from the right does not equal the limit from the left, and hence, is not differentiable in .
To prove continuity of , we want to show that . Since I haven't found any good counter-examples to this, I've tried to prove it with the epsilon-delta definition instead, with little luck.
We see that
getting me nowhere.
Trying with spherical coordinates instead, we get
I'm not sure how to proceed. Suggestions?