How to show that $f(x,y)$ is continuous. How to show that $f(x,y)$ is continuous. 
$$f(x,y)=\frac{4y^3(x^2+y^2)-(x^4+y^4)2x\alpha}{(x^2+y^2)^{\alpha +1}}$$  for $\alpha <3/2$.
Please show me   Thanks :) 

The only point where we can have trouble is $(x,y)=0$. To figure out what happens there, use the polar coordinates, as suggested in the comment of davin. We have $x=r\cos\theta$, $y=r\sin\theta$ and therefore 
$$x^2+y^2=r^2,\qquad x^4+y^4=r^4\left(\cos^4\theta+\sin^4\theta\right)$$
so that
$$f=r^{3-2\alpha}\left(4\sin^3\theta-2\alpha\cos^5\theta-2\alpha\cos\theta\sin^4\theta\right).\tag{1}$$
Obviously, as $\alpha<3/2$ and $(x,y)\rightarrow(0,0)$, the limit of $f$ in (1) exists and is equal to the value of $f$ at $(0,0)$ (equal to zero). 
