Problem

乙、計算、證明題：共 3 題，每題 12 分，共 36 分。須詳細寫出計算及證明過程，否則不予計分。

1. Let

$$f(x,y)= \begin{cases} \dfrac{(xy)^p}{x^4+y^4}, & \text{if } (x,y)\ne(0,0),\\ 0, & \text{if } (x,y)=(0,0) \end{cases}$$

Use polar coordinates to show that $f(x,y)$ is continuous at all $(x,y)$ if $p>2$, but discontinuous at $(0,0)$ if $p\le 2$.