Let $a>0$. I have to show that function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ $$f(x,y):=\frac{x^{2a}+y^{2a}}{x^2+y^2}$$ when $(x,y)\ne(0,0)$ and $f(0,0):=(0,0)$ is differentiable at $(0,0)$ if and only if $a>3/2$.
My attempt: I think it is very easy. First let's notice that this function is symmetrical in regards to both arguments. Now I check if a partial derivative with respect to $x$ exists at $(0,0)$ and if it is continuous function.
Calculate:
$$\lim_{h\rightarrow0}\frac{h^{2a}}{h^2}\cdot h^{-1}$$ It is obvious that the above limit is finite only if $a\ge3/2$. Precisely, it is equal to $1$ when $a=3/2$ and equal to $0$ if $a>3/2$.
Now, we also see that partial derivative is continuous at $(0,0)$ whenever $a>3/2$. Now we will show that $f$ is not differentiable at $(0,0)$ when $a=3/2$.
Consider a limit$$\lim_{(h,j)\rightarrow(0,0)} \frac{\frac{h^3+j^3}{h^2+j^2}-h-j}{h^2+j^2}$$. After changing to radial coordiantes, we can show that this limit depends on an angle.
Is this all ok?