Continuity, differentiability of $f(x,y)=\frac{x^{\frac{3}{2}}y^b}{x^2+y^2}$ , $x^2+y^2\ne 0 $

Question

$$f(x,y)=\begin{cases}\frac{x^{\frac{3}{2}}y^b}{x^2+y^2}&,\text{ when }x^2+y^2\ne 0 \\ 0&,\text{ when }x^2+y^2=0\end{cases}$$

Discuss $f$ ’s continuity, differentiability, partial derivative’s continuity.

My attempt:

Let $y=kx$ ,and $x\rightarrow 0$

Then $f=(1+k^2)^{-1}x^{-\frac{1}{2}+b}k^b$ then I guess I should be $>\frac{1}{2}$ and $<$ and $=$

But I don’t know how to proof the continuity ,and differentiability ,and partial derivative’s continuity on the three situation

Answer 1

Hint: Use that $$\frac{1}{x^2+y^2}\le \frac{1}{2|xy|}$$