Question

check whether the function f (x,y)={4x^2y/4x^4+y^2 , (x,y)not=to (0,0) 0 (x,y)= to (0,0). } Is continuous at (0,0) ? Justify your answer.

Answer 1

$$f(x,y) = \begin{cases} \dfrac{4x^2y}{4x^4+y^2} &\text{if } (x,y)\not=(0,0) \\ 0 &\text{if } (x,y)=(0,0) \end{cases}$$

$$\lim\limits_{(x,y)\to (0,0)}f(x,y)=[y=x^2]=\lim\limits_{(x,y)\to (0,0)}\dfrac{4x^2(x^2)}{4x^4+(x^2)^2}=$$

$$={4\over 5}\not=0=f(0,0)$$

Therefore the function $f(x,y)$ is not continuous at $(0,0).$