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Find for which values of $C \in \mathbb{R}$ the function $f$: $\mathbb{R^2} \to \mathbb{R}$ is differentiable, with $f$ defined by:

$$f(x,y) = \begin{cases} \frac{|x|^C y}{\sqrt{x^2 +y^2}} &\mbox{if } (x,y) \neq (0,0) \\ 0 &\mbox{if } (x,y) = (0,0) \end{cases}$$

I'm pretty sure I first have to find for which values of $C$ the function is continuous and study the different cases, then if it is continuous I can use the definition of the derivative to find the restrictions on $C$ for differentiability. I think I have to start by checking if $f$ is continuous at $(0,0)$ which is the only point where it might not be differentiable. Do I have the right starting idea?

Matthew Cassell
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John
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  • looks good. Keep in mind what you know about the composition of continuous functions – user190080 Nov 27 '15 at 21:18
  • I found that the function is homogeneous of degree C, since it is defined in (0,0) I can say that for C>0 it is continuous at (0,0) and for $C \leq 0$ it isn't. But is it also positive homogeneous? Because if it is I can say things about the differential in (0,0) right? – John Nov 28 '15 at 00:09

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