Continuous and differentiable points of f(x,y,z) (Existence of multivariable limit?)

The problem is stated as follows:

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Find the continuous points P and the differentiable points Q of the function in , defined as

and

.

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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'Hôpital'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?